Geometry articles, theorems, problems

What Is Geometry?
What Is Analytic Geometry?

Pictorial Geometry Index

  1. 1 + 27 = 12 + 16 Sangaku
  2. 120° Breeds 90° [Java]
  3. 3-4-5, Golden Ratio
  4. 3 Roads, 3 Travelers [Java]
  5. 3 Utilities Puzzle
  6. 3D Concurrency Of Altitudes
  7. 3D Quadrilateral - a Coffin Problem
  8. 3-4-5 Triangle by a Kid
  9. 4 Travelers problem
  10. 5-Step Construction of the Golden Ratio, One of Many
  11. 5x5 Square Grid and 5 Circes [Java, GeoGebra]
  12. 6 to 9 Point Circle [Java]
  13. 60° Angle And Importance of Being The Other End of a Diameter
  14. 60o Breeds 90o [Java]
  15. 7 = 2 + 5 Sangaku
  16. 9-point Circle as a Locus of Concurrency [Java]
  17. 9 Point Center on Angle Bisector [GeoGebra]
  18. A 60° Rhombus in a 60° Isosceles Trapezoid
  19. A better solution to a difficult sangaku problem
  20. A Case of Similarity [Java]
  21. A Chain of Touching Circles in a Polygon (a la Bui Quang Tuan) [Java]
  22. A Chain of Touching Circles in a Polygon (à la Evelyn) [Java]
  23. A Characterization of the Euler Line [Java]
  24. A Circle in an Isosceles triangle from the 2006 Itish MO
  25. A Circle Related to Incenter and Circumcenter [Java]
  26. A Circle Rolling in an Equilateral Triangle [Java]
  27. A Circle With Two Centers [Java]
  28. A Conic in a Triangle [JavaScript, GeoGebra]
  29. A Combinaton Locus
  30. A Corrected Version of the Calculus Proof of the Pythagorean Theorem
  31. A Curious Identity in Triangle
  32. A Cyclic Inequality in Triangle
  33. A Cyclic Inequality in Triangle II
  34. A Cyclic Inequality in Triangle for Integer Powers
  35. A Degenerate Case of Bottema's Configuration [Java, GeoGebra]
  36. A Diameter As a Diagonal of Inscribed Quadrilateral [Java, GeoGebra]
  37. A Discovery of Hirotaka Ebisui And Thanos Kalogerakis
  38. A dissection puzzle. Proof #99 of the Pythagorean theorem
  39. A Double Meaning of an Arc's Midpoint [Java, GeoGebra]
  40. A Family of Cyclic Quadrilaterals [JavaScript, GeoGebra]
  41. A Generalization of Simson Line [JavaScript, GeoGebra]
  42. A Generalization of the Pythagorean Theorem
  43. A Generalization of Van Aubel's Theorem
  44. A Generalized Cavalieri-Zu Principle
  45. A Geometric Limit
  46. A Geometry Problem from 2012 IMO - Problem 5
  47. A Hard but Important Sangaku
  48. A Hinged Realization of a Plane Tessellation [Java]
  49. A Lemma of Equal Areas [Java]
  50. A Lemma on the Road to Sawayama
  51. A Line in a Square Grid [Java]
  52. A Line in Triangle Through the Circumcenter [Java]
  53. A Matter of Collinearity: Metamorphosis of a Problem
  54. A Multiplicative Identity of Areas in a Triangle [Java]
  55. A Note on Cavalieri's Indivisibles
  56. A Parallelogram in Triangle [Java]
  57. A Possibly First Proof of the Concurrence of Altitudes
  58. A Problem from Greece via Peru
  59. A Problem from the 1985 Balkan Mathematical Olympiad (Shortlist)
  60. A Problem In an Isosceles Right Triangle [Java]
  61. A Problem in Pentagon with Right Angles
  62. A Problem in Pentagon with Supplementary Angles
  63. A Problem in Three Pentagons [JavaScript, GeoGebra]
  64. A Problem of Acute Triangle
  65. A Problem of Two Incircles by Miguel Ochoa Sanchez
  66. A Problem on an Icy Cone
  67. A problem with equilateral triangles [Java]
  68. A Problem with Right Isosceles Triangles [JavaScript]
  69. A Problem with Two Isosceles Triangles [JavaScript]
  70. A Problem with Two Similar Rectangles
  71. A Proof Perigal and All Others After Him Missed
  72. A Property of a Median [JavaScript, GeoGebra]
  73. A Property of Circle Through the Incenter [JavaScript]
  74. A Property of Cyclic Quadrilateral
  75. A Property of Equiangular Polygons [Java]
  76. A Property of Inscriptible Quadrilaterals [JavaScript]
  77. A Property of Isogonal Lines [JavaScript, GeoGebra]
  78. A Property of Orthic Triangle [JavaScript, GeoGebra]
  79. A Property of Pascal's Hexagon Pascal May Have Overlooked
  80. A Property of Perspective Triangles [JavaScript, GeoGebra]
  81. A Property of Points on Excircle
  82. A Property of Points on Incircle
  83. A Property of Rhombi [Java]
  84. A Property of the Line IO [Java]
  85. A Property of the Line IO: A Bundled Statement and Proof [Java]
  86. A Property of the Line IO: a proof from the Book [Java]
  87. A Property of Trapezoid with a Right Angle
  88. A Relation in Inscribed Tetrahedra
  89. A Relation in Triangle
  90. A Restored Sangaku
  91. A Sangaku Follow-Up on an Archimedes' Lemma [Java]
  92. A Sangaku: Two Unrelated Circles [Java]
  93. A Sangaku with an Egyptian Attachment [Java]
  94. A Sangaku with Many Circles and Some [Java]
  95. A Simple Solution to a Difficult Sangaku Problem
  96. A Special Point: Intersection of the Diagonals in Quadrilateral
  97. A Special Triangle with a Line Through the Lemoine Point
  98. A square in parallel lines
  99. A Stronger Triangle Inequality [Java]
  100. A Sushi Morsel
  101. A Three Pegs Question [Java]
  102. A Triangle of Antreas Hatzipolakis [Java]
  103. A Triangle With a 45 Degrees Angle in Square
  104. A Trigonometric Identity in a Right Triangle from the 1960 IMO
  105. A Trigonometric Solution to a Difficult Sangaku Problem
  106. A Two-Triangle Inequality
  107. A Two-Triangle Inequality II
  108. A View on the Circle of Apollonius [JavaScript,GeoGebra]
  109. About a Line and a Triangle [Java]
  110. Abul Wafa al-Buzjani's Proof of the Pythagorean Theorem [Java]
  111. Abutting Equilateral Triangles on a Line I [JavaScript, GeoGebra]
  112. Acute Angle Bisectors in a Right Triangle [Java]
  113. Adams' Circle [Java]
  114. Addition and Subtraction Formulas for Sine and Cosine
  115. Addition and Subtraction Formulas for Sine and Cosine II
  116. Adil Abdulayev's Inequality With Angles, Medians, Inradius and Circumradius
  117. Adil Abdullayev's Inequality With Roots and Powers
  118. Algebraic proof of the theorem of butterflies in quadrilaterals
  119. Algebraic-Geometric Equation
  120. All about altitudes
  121. All Triangles Are Isosceles [Java]
  122. All Trigonometric Inequality in Triangle
  123. Altitudes and the Euler Line [GeoGebra]
  124. Altitudes and the Power of a Point [Java]
  125. Altitudes Concur at a Point: a Second Look
  126. Altitudes have ears, foot, stem, and root
  127. An Affine Property of Barycenter
  128. An Algebraic Lemma with Geometric Consequences [JavaScript]
  129. An All-Inclusive Inequality
  130. An All-Inclusive Inequality II
  131. An Angle in Square Conjoint with Regular Pentagon
  132. An Angle Inequality in Simple Polygons
  133. An Area Inequality
  134. An Area Inequality in Right Triangle
  135. An Asymmetric Inequality
  136. An Euclidean Construction with Inversion [Java, GeoGebra]
  137. An Extension of van Schooten's Theorem
  138. An Extra Triple of Equilateral Triangles for Napoleon [JavaScript, GeoGebra]
  139. An Identity for Regular Tetrahedra
  140. An Identity in a Regular Heptagon
  141. An Identity in (Cyclic) Quadrilaterals [JavaScript, GeoGebra]
  142. An Identity in Cyclic Quadrilateral
  143. An Identity in Triangle
  144. An Identity in Triangle II
  145. An Identity in Triangle with a 135 Degrees Angle
  146. An Inequality in Triangle with a Constraint
  147. An Inequality in Triangle with Side Lengths and Circumradius
  148. An Inequality with Exradii and an Altitude
  149. An Identity with Inradius and Circumradii
  150. An Inequality for Sides and Area
  151. An Inequality for the Cevians through Circumcenter
  152. An Inequality for the Cevians through Spieker Point via Brocard Angle
  153. An Inequality from the 2015 Romanian TST
  154. A Cyclic Inequality from the 6th IMO, 1964
  155. An Inequality in Acute Triangle, Courtesy of Ceva's Theorem
  156. An Inequality in Cyclic Quadrilateral
  157. An Inequality in Cyclic Quadrilateral II
  158. An Inequality in Cyclic Quadrilateral III
  159. An Inequality in Cyclic Quadrilateral IV
  160. An Inequality of the Areas of Triangles Formed by Circumcenter And Orthocenter
  161. An Inequality on Circumscribed Quadrilateral
  162. An Inequality in Triangle
  163. An Inequality in Triangle II
  164. An Inequality in Triangle III
  165. An Inequality in Triangle IV
  166. An Inequality in Triangle, V
  167. An Inequality in Triangle, VI
  168. An Inequality in Triangle XI
  169. An Inequality in Triangle, Mostly with the Medians
  170. An Inequality In Triangle That Involves the Four Basic Centers
  171. An Inequality in Triangle with Altitudes, Medians And Symmedians
  172. An Inequality in Triangle, with Integrals
  173. An Inequality in Triangle with Roots and Circumradius
  174. An Inequality in Triangle, with Sines
  175. An Inequality in Triangle, with Sines II
  176. An Inequality in Triangle with the Circumradius, Inradius and Angle Bisectors
  177. An Inequality in Triangle with the Sines of Half-Angles and Cube Roots
  178. An Inequality Involving Fermat Point
  179. An Inequality Not in Triangle
  180. An Inequality of Degree 3 with Inradius
  181. An Inequality with Altitudes and Angle Bisectors
  182. An Inequality with Altitudes and Medians

    An Inequality with Angle Trisectors

  183. An Inequality with Arctangents in Triangle
  184. An Inequality with Circumradii And Distances to the Vertices
  185. An inequality with Cosines and a Sine
  186. An Inequality with Inradius and Side Lengths
  187. An Inequality with Cotangents And the Circumradius
  188. An Inequality with Tangents and Sides
  189. An Inequality with One Tangent and Six Sines
  190. An inequality with Three Points
  191. An Inequality in Triangle with Differences of the Medians
  192. An Inequality with Inradius and Circumradii
  193. An Inequality in Triangle with the Medians, Sines and Cube Roots
  194. An Inequality with Powers of Six
  195. An Inequality with Sides, Altitudes, Angle Bisectors and Medians
  196. An Inequality with Sides, Cosines, and Semiperimeter
  197. An Inequality with Sin, Cos, Tan, Cot, and Some
  198. An Inequality with Sines
  199. An Inequality with Tangents and Cotangents
  200. An Inequality with the Most Important Cevians
  201. An Inradii Relation in Inscriptible Quadrilateral [JavaScript]
  202. An Isoperimetric Theorem [Java]
  203. An Old Japanese Theorem
  204. An Unexpected Pair of Similar Triangles Which Are Equal [JavaScript, GeoGebra]
  205. An Unexpected Rhombus [JavaScript, GeoGebra]
  206. An Universal Inequality for Cevians
  207. An Unusual Generalization of the Pythagorean Theorem
  208. An Unusual Generalization of the Pythagorean Theorem, a Variant
  209. Angle: An Illustrated Classification [Java]
  210. Angle Bisector in Square
  211. Angle Bisectors Cross Circumcircle
  212. Angle Bisector in Equilateral Trapezoid
  213. Angle Bisector in Parallelogram [Java]
  214. Angle Bisectors In Rectangle [Java]
  215. Angle Bisector in Touching Circles [Java]
  216. Angle Bisectors [Java]
  217. Angle Bisectors and Perpendiculars in a Quadrilateral [Java]
  218. Angle Bisectors in Ellipse [Java]
  219. Angle Bisectors in Ellipse II [Java]
  220. Angle Bisectors in a Quadrilateral [Java]
  221. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise [Java]
  222. Angle Bisectors On Circumcircle [Java]
  223. Angle Chasing on Radical Axis [JacaScritp, GeoGebra]
  224. Angle Chasing in a 75-30-75 triangle
  225. Angle in Square
  226. Angle in Right Triangle
  227. Angle Preservation Property [Java]
  228. Angle Subtended by a Diameter [Java]
  229. Angle Trisection, the Impossibility of Thereof
  230. Angle Trisection by Archimedes [Java]
  231. Angle Trisection by Hippocrates [Java]
  232. Angle Trisection by Paul Vjecsner [Java]
  233. Angle Trisectors on Circumcircle [Java]
  234. Angle Trisection by Paper Folding
  235. Angles in 4-5-6 Triangle [JavaScript]
  236. Angles in a Cube I [Java]
  237. Angles in a Cube II [Java]
  238. Angles in a Cube III [Java]
  239. Angles in Triangle Add to 180o
  240. Angles in a Triangle: an Exercise
  241. Angles Inscribed in an Absent Circle [Java]
  242. Another Concurrence on the 9-Point Circle [JavaScript, GeoGebra]
  243. Another Equilateral Triangle in Napoleon's Configuration [JavaScript, GeoGebra]
  244. Another Geometry Problem from 2012 IMO - Problem 1
  245. Another Golden Ratio in Semicircle
  246. Another Identity in Triangle
  247. Another Pair of Archimedean Twins - a Redux [JavaScript, GeoGebra]
  248. Another Property of Points on Incircle
  249. Another Property of the 9-Point Circle [JavaScript, GeoGebra]
  250. Another Refinement of the Ionescu-Weitzenbock Inequality
  251. Another Sangaku in Square
  252. Another Seven Circles Theorem [JavaScript, GeoGebra]
  253. Anticenter and Orthocenters [JavaScript, GeoGebra]
  254. Anticomplementary Triangle Surprise [Java]
  255. Antiparallel [Java]
  256. Antiparallel and Circumradius [JavaSCript, GeoGebra]
  257. Antiparallel via Three Reflections [Java]
  258. APMO 1995 Problem 3
  259. APMO 1998, Problem 4
  260. APMO 2002, Problem 3
  261. APMO 2003, Problem 2
  262. APMO 2004, Problem 2
  263. The Averages of Sides and Angles in a Triangle
  264. The Problem of Apollonius
  265. Approximate Construction of Regular Pentagon by A. Dürer
  266. Arbelos -- the Shoemaker's Knife [Java]
  267. Another Pair of Twins in an Arbelos [Java]
  268. Archimedes' Quadruplets [Java]
  269. Archimedes' Twin Circles and a Brother [Java]
  270. Area and Perimeter of Circle
  271. Area Inequalities in Triangle
  272. Area Inequality in Three Triangles
  273. Area of an Arbelos [Java]
  274. Area of Cevian Triangle
  275. Area of Cutoff Triangles
  276. Area of Rectangle
  277. Area of the Intersection of Two Equal Rectangles [JavaScript, GeoGebra]
  278. Area of Trapezoid
  279. Area Optimization in Trapezoid
  280. Area, Sector Area, and Segment Area of an Ellipse
  281. Areas between Two Parallelograms [JavaScript, GeoGebra]
  282. Areas in Three Squares [JavaScript, GeoGebra]
  283. Average Distance to N Points
  284. Chain of Circles on a Chord [Java]
  285. Chain of Four Intersecting Circles [JavaScript, GeoGebra]
  286. Chain of Inscribed Circles [Java]
  287. Chain of Six Intersecting Circles [JavaScript, GeoGebra]
  288. Characterization of the Symmedian Point with Medians and Orthic Triangle
  289. Characterization of Trapezoid [JavaScript, GeoGebra]
  290. Chasing Secant Angles [JavaScript]
  291. Concurrence in Arbelos [JavaScript, GeoGebra]
  292. Concurrency in Arbelos [Java]
  293. Concurrence on Angle Bisector [JavaScript, GeoGebra]
  294. Concurrent Chords in a Circle, Equally Inclined
  295. Concyclic Points in Arbelos [Java]
  296. Rectangle in Arbelos [Java]
  297. Rectangular Areas in Circle [JavaScript, GeoGebra]
  298. Red and Pink Areas
  299. Reuleaux's Triangle, Extended [GeoGebra]
  300. Small Triangle from Small Triangle
  301. Square, 45 Degrees Angle and Pythagoras' Look-alike
  302. Square, Semicircle, and Other Curves [JavaScript, GeoGebra]
  303. Squares in Arbelos [Java]
  304. Twin Segments in Arbelos [Java]
  305. Archimedean Siblings out of Wedlock, i.e., Arbelos (JavaScript, GeoGebra)
  306. Archimedes'Book of Lemmas
  307. Arbelos' Morsels [JavaScript, GeoGebra]
  308. Archimedes' Law of the Lever [Java]
  309. Archimedes' Method
  310. Archimedes' Twins in the Edo Period
  311. Area and Perimeter Splitters in a Triangle
  312. Area Inequality in Trapezoid
  313. Area Inequality in Triangle
  314. Area Inequality in Triangle II
  315. Area of a Bicentric Quadrilateral
  316. Area of a Circle by Leonardo da Vinci [Java]
  317. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
  318. Area of Isosceles Triangle
  319. Area of Median Triangle [Java]
  320. Area of Parallelogram [Java]
  321. Area of Parallelogram Formula by Shearing
  322. Area of the Union of Two Squares [Java]
  323. Area of Triangle [Java]
  324. Areas and Centroid in a Triangle [Java]
  325. Areas In Circle
  326. Areas in Square by Dissection [Java]
  327. Areas in Two Parallelograms and a Translation [JavaScript, GeoGebra]
  328. Arithmetic-Geometric Mean Inequality
  329. Arithmetic Mean Sangaku [Java]
  330. Arithmetic Mean Sangaku II [Java]
  331. Around the Incircle [Java]
  332. Around the Incircle in a Polygon [Java]
  333. Artisans vs Mathematicians
  334. Asian Pacific Kite [Java]
  335. Assigning Numbers to Points in the Plane
  336. Assimilation Illusion [Java]
  337. Axis of Similitude of Three Circles
  338. Barbier, The Theorem of [Java]
  339. Barycenter of Cevian Triangle
  340. Barycentric coordinates
  341. Batman's Problem [JavaScript, GeoGebra]
  342. Bender: A Visual Illusion [Java]
  343. Bespectacled Eyeballs Extension [JavaSCript, GeoGebra]
  344. Between a Cross and a Square (lies a Golden Ratio)
  345. Between Major and Minor Circles [Java]
  346. Bevan's Point and Theorem [Java]
  347. Beyond Pythagoras
  348. Beyond Viviani [JavaScript]
  349. Bimedians and Diagonals in a Quadrilateral
  350. Bimedians in a Regular Tetrahedron [Java]
  351. Bisectal Circle [Java]
  352. Bisecting arcs
  353. Bisecting a shape
  354. Bisection of Yin and Yang
  355. Bisector of an imaginary angle may be real
  356. Bisectors and Proportions [JavaScript]
  357. BMO 2014, Problem 3 [JavaScript, GeoGebra]
  358. Bottema Shatters Japan's Seclusion [Java]
  359. Bottema's theorem [Java]
  360. Bottles in a Wine Rack [Java]
  361. Bounded Distance
  362. Bounds for the Sum of Distances to the Vertices [JavaScript]
  363. Brahmagupta's Formula and Theorem
  364. Brahmagupta's Theorem [Java]
  365. Brahmagupta-Mahavira Identities
  366. Brianchon in Ellipse [Java]
  367. Brianchon's theorem [Java]
  368. Brick Wall Illusion [Java]
  369. Bride's Chair [Java]
  370. Brief introduction to Rational Trigonometry
  371. Brocard Point and a Relation of Circumradii
  372. Broken Chord Theorem [Java]
  373. Broken Line in Triangle
  374. Building a Bridge [Java]
  375. Building Bridges [Java]
  376. Bulging lines illusion [Java]
  377. Butterfly Theorem
  378. Cabart's Collinearity
  379. Camouflaged Butterfly
  380. Cantor Set and Function
  381. Cantor's Theorem [Java]
  382. Carnot's Theorem
  383. Carnot's Theorem
  384. Carnot's Theorem from Ptolemy's Theorem
  385. Carnot's Theorem (Generalization of Wallace's theorem) [Java]
  386. Carnot's Theorem for Conics
  387. Carpet With a Hole [Java]
  388. Carpets Theorem [Java]
  389. Casey's Theorem [Java]
  390. Cassini's Ovals and Geometric Optimization [JavaScript, GeoGebra]
  391. centers of similitude
  392. Centroid, a Characteristic Property Of
  393. Centroid and Circumcenter in Isosceles Tetrahedra
  394. Centroid and Incenter in Isosceles Tetrahedra
  395. Centroid on the Incircle in Isosceles Triangle
  396. Centroid on the Incircle in Right Triangle
  397. Centroids and Circumcenters
  398. Centroids in a Polygon [Java]
  399. Cevian Parallelogram [JavScript, GeoGebra]
  400. Chasing Angles Among Angle Bisectors [JavaScript]
  401. Chasing Angles in Pascal's Hexagon [Java]
  402. Circle, Isosceles Triangle and Fixed Point [Java]
  403. Circle Centers on Radical Axes [Java]
  404. Circle Chains on Napoleon Triangles [Java]
  405. Circle Concurrence on Circumcircle/a> [JavaScript, GeoGebra]
  406. Circle in a Square Inscribed in a Circle [Java]
  407. Circle Rotations and Fixed Points
  408. Circles in Barycentric Coordinates
  409. Circles on the sides of a triangle from the 2005 Austrian MO
  410. Circles Tangent to Circumcircle [Java]
  411. Circles Tangent to Medians at the Centroid [JavaScript, GeoGebra]
  412. Circles with Tangent Diameters [Java]
  413. Ceva and Menelaus Meet on the Roads
  414. Ceva in Circumscribed Quadrilateral
  415. Ceva's Theorem
  416. Ceva's Theorem and Fibonacci Bamboozlement [Java]
  417. Cevian Cradle [Java]
  418. Cevian Cradle II [JavaScript, GeoGebra]
  419. Cevian Nest [Java]
  420. Cevian Triangle [Java]
  421. Cevians And Semicircles [JavaScript]
  422. Cevians Through the Circumcenter
  423. Chain of Inscribed Circles [Java]
  424. Chaos, Emergence of [Java]
  425. Characteristic Property Of Right Triangles
  426. Chasles' Theorem
  427. Cherchez le quadrilatère cyclique [Java]
  428. Cherchez le quadrilatère cyclique II [Java]
  429. Chords, Concurrency and Orthic Triangle [Java]
  430. Chromatic Number of the Plane
  431. Chvatal's Art Gallery Theorem [Java]
  432. Circle and Hyperbola as Lighthouse Curves
  433. Circle and Hyperbola as Lighthouse Curves II
  434. Circle Crosses Triangle [Java]
  435. Circle Concurrency and Spiral Similarity [Java]
  436. Circle Division by Euclid and Leonardo [Java]
  437. Circle of Apollonius in Equilateral Triangle
  438. Circle of Similitude [Java]
  439. Circle Rotations and Fixed Points [JavaScript, GeoGebra]
  440. Circle through the Circumcenter [Java]
  441. Circle through the Incenter [Java]
  442. Circle through the Incenter And Antiparallels [Java]
  443. Circle-Stacking Theorem [Java]
  444. Circles And Parallels [Java]
  445. Circles and Semicircles in Rectangle [Java]
  446. Circles Cover a Quadrilateral [Java]
  447. Circles in a Circular Segment
  448. Circles in a Regular Polygon [Java]
  449. Circles in Morley's Triangles [Java]
  450. Circles Lined on the Legs of a Right Triangle [Java]
  451. Circles On Cevians [Java]
  452. Circles through the Orthocenter [Java]
  453. Circles with Equal Collinear Chords [Java]
  454. Circular Poggendorff Illusion [Java]
  455. Circle Inscribed in a Circular Segment [Java]
  456. Circumcenter and Orthocenter
  457. Circumcenter and Orthocenter Are Isogonal Conjugate
  458. Circumcenter on Angle Bisector [Java]
  459. Circumcevian Triangle [Java]
  460. Classification of Quadrilaterals [Java]
  461. Cleavance Center [Java]
  462. Cleaver [Java]
  463. Clifford Algebras
  464. Clifford's Chain [Java]
  465. Climbing Pyramidal Slopes
  466. Clough's theorem: the simplest proof [JavaScript, GeoGebra]
  467. Coaxal Circles on Perpendicular Bisector [Java]
  468. Coaxal Circles Theorem [Java]
  469. Collage Theorem [Java]
  470. Collinear Intersections and Products of Ratios [JavaScript, GeoGebra]
  471. Collinearity from the IMO 2013 (Problem 4) [Javacript, GeoGebra]
  472. Collinearity in Bicentric Quadrilaterals [Java]
  473. Collinearity in Tangent Circles
  474. Collinearity via Concyclicity [Java]
  475. Collinearity with the Orthocenter [Java]
  476. Color Cycling on the Mandelbrot Set [Java]
  477. Coloring Plane with Three Colors
  478. Coloring Points in the Plane
  479. Common Centroids Lead to Equilateral Triangle [JavaScript, GeoGebra]
  480. Common Chord and a Tangent [Java]
  481. Common Tangents to Two Circles I [Java]
  482. Common Tangents to Two Circles II [Java]
  483. Compass Only Construction - A Chord Tangent to an Inner circle [Java, GeoGebra]
  484. Complete Quadrilateral [Java]
  485. Complex Numbers and Geometry
  486. >Concurrence in Right Triangle
  487. Concurrence on a Circle [JavaScript, GeoGebra]
  488. Concurrence Not from School Geometry [Java, GeoGebra]
  489. Concurrency of Medians and Medial Triangles
  490. Concurrency of the Altitudes in a Triangle - Trigonometric Proof
  491. Concyclic Points at the IMO 2014 [JavaScript, GeoGebra[
  492. Concyclicity in Rectangle [Java]
  493. Concurrence in Equilateral Triangle [JavaScript,GeoGebra]
  494. Concurrence of Ten Nine-Point Circles [JavScript, GeoGebra]
  495. Concurrency in the Intouch Triangle [Java]
  496. Concurrent and Parallel Lines in Parallelogram [JavaScript, GeoGebra]
  497. Concurrent Cevians and a Conic through Their Feet [JavaScript, GeoGebra]
  498. Concyclic Points from Midpoint of an Arc [GeoGebra, JavaScript]
  499. Concyclic Incenters in Bicentric Quadrilateral [JavaScript, GeoGebra]
  500. Concyclic Points in Inscriptible Quadrilateral [JavaScript, GeoGebra]
  501. Concyclicity from Collinearity [JavaScript,GeoGebra]
  502. Concyclicity in Two Semicircles [JavaScript, GeoGebra]
  503. Concyclic Points of Two Ellipses with Orthogonal Axes [JavaScript, GeoGebra]
  504. Condition to Have a 60 Degrees Angle
  505. Concyclic Circumcenters: A Dynamic View [Java]
  506. Concyclic Points in a Triangle
  507. Concyclic Points in Bride's Chair [Java]
  508. Concyclic Points in Equilateral Bumps [Java]
  509. Conic from Parallel Chords [JavaScript, GeoGebra]
  510. Conic in Hexagon [Javascript, GeoGebra]
  511. Conic in Mixtilinear Incircles [JavaScript, GeoGebra]
  512. Conic Sections
  513. Conic Sections as Loci of Points [Java]
  514. Conics Intersecting Sides Of A Triangle [JavaScript, GeoGebra]
  515. Conics in Triangle [JavaScript, GeoGebra]
  516. Conics Related To In- and Excircles [JavaScript, GeoGebra]
  517. Conjugate Diameters in Ellipse [Java]
  518. Consecutive Isosceles Decomposition [Java]
  519. Constant Chord
  520. Constrained Area in Triangle
  521. Constant Ratio on Circle [JavaScript, GeoGebra]
  522. Construct Triangle from Angle, Inradius, and Altitude [JavaScript, GeoGebra]
  523. Construct Triangle by Angle, Median, and Circumradius [JavaScript, GeoGebra]
  524. Construct Triangle by Angle Bisector, Altitude, and Side [JavaScript, GeoGebra]
  525. Construct Triangle by Angle, Altitude and Median [JavaScript, GeoGebra]
  526. Construct Triangle from Altitude, Median and Inradius [JavaScript, GeoGebra]
  527. Constructing a triangle from its angle bisectors is in general impossible
  528. Construction: Parallel Line through Point
  529. Construction and Properties of Mixtilinear Incircles [JavaScript, GeoGebra]
  530. Construction and Properties of Mixtilinear Incircles 2 [JavaScript, GeoGebra]
  531. Construction of a Cyclic Quadrilateral
  532. Construction of a Square from the 2008 Austrian MO
  533. Construction of a Triangle from an Angle, Circumcenter and Incenter
  534. Construction of a Triangle from the Circumcenter, Orthocenter and Incenter
  535. Construction of a Triangle from the Circumradius, Inradius, and an Altitude
  536. Construction of a Triangle from a Side, the Median and the Angle Bisector
  537. Construction of a Triangle from Two Vertices and the Centroid
  538. Construction of an Oval - the Roman Way [Java, GeoGebra]
  539. Construction of Conics from Pascal's Theorem [Java]
  540. Construction of Regular Pentagon by H. W. Richmond
  541. Construction of Thebault Circles [JavaScript, GeoGebra]
  542. Construction of the Angle Bisector [Java]
  543. Construction of the Perpendicular Bisector [Java]
  544. Construction of Triangle from Side, Angle, and that Angle's Bisector
  545. Construction problem [Java]
  546. Constructions Related To An Inaccessible Point
  547. Construction with the Compass Only
  548. Continuous Line Illusion [Java]
  549. Convex Polygon Is the Intersection of Half Planes [Java]
  550. Copernicus' Theorem [Java]
  551. Construction with Optimization in Angle - Problem the 1979 USA Mathematical Olympiad
  552. cos(36°) = (1 +5)/4
  553. cos 75 + cos 45 = cos 15
  554. Cosine Law
  555. Counting Triangles and Segments in a Polygon [Java]
  556. Cross Points in a Polygon [Java]
  557. Cross-Ratio
  558. Crossed-Lines Construction of Shapes of Constant Width [Java]
  559. Crossing Number of a Graph
  560. Cube Counting Ambiguity [Java]
  561. Cultivating Regular Pentagons
  562. Curious Irrationality in Square
  563. Curry's Paradox [Java]
  564. Cut the Cone [Java]
  565. Cut the Cube [Java]
  566. Cutting a Hole in Half [JavaScript, GeoGebra]
  567. Cut the Cylinder [Java]
  568. Cutting Circle to Rearrange Two Dots [JavaScript]
  569. Cutting Squares
  570. Cutting Triangle in Two by a Line through a Point [Java, GeoGebra]
  571. Cutting Triangles off Regular Hexagon
  572. Cyclic Hexagon [Java]
  573. Cyclic Quadrilateral, Angle Bisector And Isosceles Triangle
  574. Cyclic Quadrilateral, Concurrent Circles and Collinear Points [Java]
  575. Cyclic Quadrilateral from the USAMO
  576. Cycloids [Java]
  577. Dan Sitaru's Inequality with Tangents II
  578. Dancing Rectangles Model Auxetic Behavior [Java]
  579. Dancing Squares or a Hinged Plane Tessellation [Java]
  580. Dao's Archimedean Twins [JavaScript, GeoGebra]
  581. Dao's Archimedean Twins - Second Pair [JavaScript]
  582. Dao's Archimedean Twins - Third Pair [JavaScript]
  583. Dao Thanh Oai's Invariant
  584. Dao's Variant of Thébault's First Problem [JavaScript, GeoGebra]
  585. David Richeson's Extention of an Old Japanses Theorem [Java]
  586. Day's Sine Illusion [Java]
  587. Decompose Cyclic Quadrilaterals into k≥4 Cyclic Quadrilaterals
  588. A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
  589. Delboeuf Illusion [Java]
  590. Delian Problem Solved [Java]
  591. Desargues' Hexagon [Java, GeoGebra]
  592. Desargues in the Bride's Chair (with Pythagoras)
  593. Desargues' Theorem [Java]
  594. Determination ofπ
  595. Diagonal Count [Java]
  596. Diagonals in a Cyclic Quadrilateral
  597. Diameters and Chords [Java]
  598. Dimensionless Inequality in the Euclidean Plane
  599. Directly Similar Figures [Java]
  600. Disappearing Lines puzzle [Java]
  601. Discovery of Duane DeTemple [Java]
  602. Dissection of a 10×13 Rectangle into Two Chessboards
  603. Dissection of a Vase [Java]
  604. Dissection of Cyclic Quadrilateral [Java]
  605. Dissection of Triangle into Rhombus [Java]
  606. Dissection of Triangle into Rhombus by Hubert Shutrick [Java]
  607. Distance
  608. Distance Between Projections
  609. Distance between the Orthocenter and Circumcenter
  610. Distance Formula
  611. Distorted Lines Illusion [Java]
  612. Divergent Spirals Illusion [Java]
  613. Divide a Circle into N Parts of Equal Area
  614. Divide a segment into n equal parts
  615. Divide Triangle by Lines Parallel to Base [JavaScript, GeoGebra]
  616. Dividing Evenly a Quadrilateral [Java]
  617. Dividing Evenly a Quadrilateral II [Java]
  618. Dorin Marghidanu's Inequality with Maximum Side
  619. Double Generation Theorem [Java]
  620. Drawing circles [Java]
  621. Droz-Farny Circles [Java]
  622. Droz-Farny Line Theorem [Java]
  623. Droz-Farny Theorem - an Analytic Solution [JavaScript, GeoGebra]
  624. Dual of Maxwell's Theorem
  625. Duplication of the Cube
  626. Dynamic Construction of Ellipse and Other Curves [Java]
  627. Dynamic Software as Serendipity Enhancement [Java, GeoGebra]
  628. A. G. Samosvat's Proof by Dissection of the Pythagorean Theorem (#104) [GeoGebra]
  629. E. W. Dijkstra's Proof of the Pythagorean Theorem [Java]
  630. Early Refinement of the Ionescu-Weitzenbock Inequality
  631. Easy Construction of Bicentric Quadrilateral [Java]
  632. Easy Construction of Bicentric Quadrilateral II [Java]
  633. Egyptian Triangle By Paper Folding
  634. Egyptian Triangle By Paper Folding II
  635. Egyptian Triangle By Paper Folding III
  636. Eight Equal Tangents [Java]
  637. Eight Point Circle [Java]
  638. Elementary Introduction into the Concept of Area
  639. Ellipse Between Two Circles [Java]
  640. Ellipse in Arbelos [Java]
  641. Ellipse Touching Sides of Triangle at Midpoints [Java, GeoGebra]
  642. Emil Stoyanov's From Reflections to Altitudes
  643. Emil Stoyanov's New Year's Problem
  644. Emil Stoyanov's New Year's Problem, a Sequel
  645. Equal Angles in Two Circles [Java]
  646. Equal Angles in Two Circles II [Java]
  647. Equal areas In Circle [JavaScript]
  648. Equal Areas in Equilateral Triangle [Java]
  649. Equal Areas In Parallelogram à la Pythagoras [Java]
  650. Equal Areas in Regular 2n-gons [Java]
  651. Equal Chords in Crossing Circles [Java]
  652. Equal Circles, Medial Triangle and Orthocenter [Java]
  653. Equal Incircles Theorem [Java]
  654. Equiangular p-gons
  655. Equichordal points: Just Do It [Java]
  656. Equidecomposition of a Rectangle and a Square [Java]
  657. Equidecomposition of Two Rectangles [Java]
  658. Equidecomposition of a Triangle and a Rectangle [Java]
  659. Equidecomposition of a Triangle and a Rectangle II [Java]
  660. Equidecomposition of Two Parallelograms [Java]
  661. Equilateral and 3-4-5 Triangles [Java]
  662. Equilateral triangle and squares
  663. Equilateral Triangle in Equilateral Triangle [JavaScript, GeoGebra]
  664. Equilateral Triangle on a Closed Curve [JavaScript, GeoGebra]
  665. Equilateral Triangle on Angle Bisectors
  666. Equilateral Triangle on Parallel Lines [Java]
  667. Equilateral Triangle on Parallel Lines II
  668. Equilateral Triangle on Three Lines [JavaScript, GeoGebra]
  669. Equilateral Triangle in Square and Its Circumcircle
  670. Equilateral Triangle in Square with the Cevians through Its Apex
  671. Equilateral Triangle in Square And the Pedal Circle of Its Apex
  672. Equilateral Triangle, Straight Line and Tangent Circles [Java]
  673. Equilateral Triangles and Incircles in a Square
  674. Equilateral Triangles formed by Circumcenters [JavaScript, GeoGebra]
  675. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral [Java]
  676. Equilateral Triangles On Sides of a Parallelogram [Java]
  677. Equilateral Triangles On Sides of a Parallelogram II [Java]
  678. Equilateral Triangles on Sides of a Parallelogram III [JavaScript, GeoGebra]
  679. Equilateral Triangles on Sides of a Quadrilateral [Java]
  680. Equilic Quadrilateral I, a Flawed Proof [Java]
  681. Equilic Quadrilateral I, A Variation [Java]
  682. Equilic Quadrilateral II [Java]
  683. Erdös-Mordell Inequality
  684. Escher's Theorem [Java]
  685. Euclid's Fifth Postulate
  686. Euclid'sElements Reference Page
  687. Euclid I.43 [Java]
  688. Euclid I.43 Extended [Java]
  689. Euclidean Construction of Center of Ellipse [Java]
  690. Euclidean Construction of Tangent to Ellipse
  691. Euler Inequality in Triangle
  692. Euler Line Cuts Off Equilateral Triangle [Java]
  693. Euler Line and 9-Point Circle [Java]
  694. Euler's Formula and Poncelet Porism
  695. Euler Line in a Triangle with a 60 Degrees Angle
  696. Every Parallelogram Is a Rectangle
  697. Excircle in Equilateral Triangle
  698. Excircles in Trapezoid
  699. Exeter Point [Java, GeoGebra]
  700. Existence of the Circumcenter [Java]
  701. Existence of the Orthocenter [GeoGebra]
  702. Existence of the Euler Line: An Elementary Proof [Java]
  703. Existence of the Incenter: a Second Look
  704. Experimentation with Dynamic Geometry Software: An Example [Java]
  705. Exploring Vecten's Wreaths with Linear Algebra
  706. Expressions Invariant under Projecive Mappings
  707. Exterior Angle Theorem - an appreciation
  708. External Angle Bisector [Java]
  709. Extouch Triangle in Poncelet Porism [Java]
  710. Extra Feature of Van Aubel Configuration [JavaScript,GeoGebra]
  711. Extras in Bottema's Configuration [JavaScritpt, GeoGebra]
  712. Extremal Problem in a Quadrilateral
  713. Extremal Distance Ratio [GeoGebra]
  714. Extremal Problem in a Circular Segment
  715. The Extreme Principle
  716. Eye-to-Eye Theorem I [JavaScript, GeoGebra]
  717. Eye-to-Eye Theorem II [JavaScript, GeoGebra]
  718. Eyeball Theorem [JavaScript, GeoGebra]
  719. Eyeball Theorem Rectified [JavaScript]
  720. Eyeballing a Ball [JavaScript, GeoGebra]
  721. Eyeballs Projected [JavaScript, GeoGebra]
  722. Fagnano's problem [Java]
  723. Faulty Symmetry
  724. Fermat's Hexagon [JavaScript, GeoGebra]
  725. Fermat in a Triangle with a 60 Degrees Angle
  726. Fermat point and 9-point Centers [Java]
  727. Fermat Point Several Times Over
  728. Feuerbach's Theorem: A Proof [Java]
  729. Find a Common Chord of Given Length [Java]
  730. Find the Angle! [Java]
  731. Finding a Parallelogram in 3D
  732. Finding Circle Center By Ruler Alone [Java]
  733. Five Concyclic Points [Java]
  734. Five Incircles in a Square [Java]
  735. Five Incircles Theorem [Java]
  736. Five Squares in Complex Numbers [Java, GeoGebra]
  737. Fixed Point in a Kite - Problem from the 1993 Asian Pacific MO
  738. Fixed Point in Isosceles and Equilateral Triangles
  739. Fixed Point on Perpendicular Bisector [GeoGebra, JavaScript]
  740. Fixed Point of Circles Orthogonal to the Given One [Java]
  741. Fixed points from 2007 Irish Mathematical Olympiad
  742. Fleury's Algorithm and Euler's Paths and Cycles [Java]
  743. FJG Capitan's Sangaku
  744. Focal Definition of Ellipse [Java]
  745. Focal Definition of Parabola [Java, GeoGebra]
  746. Focus and Directrix of Ellipse [Java]
  747. Focus on the Eyeball Theorem [JavaScript, GeoGebra]
  748. Folding Square in a Line through the Center [Java, GeoGebra]
  749. Fold Square into Equilateral Triangle [JavaScript]
  750. Following the Hilbert Curve [Java]
  751. Foot of Altitude and Minimum Distance [Java]
  752. For Equality Choose Angle Bisector [JavaScript, GeoGebra]
  753. For every object there is a distance at which it looks its best.
  754. Ford's touching circles
  755. Four Centroids and Parallels [Java]
  756. Four Circles and a Line [GeoGebra]
  757. Four Circles In a Triangle [Java]
  758. Four Concurrent Lines in a Cyclic Quadrilateral [Java]
  759. Four Construction Problems [Java]
  760. Four Crossing Circles [Java]
  761. Four Geometric Problems of Antiquity
  762. Four Golden Circles
  763. Four Hinged Squares [Java]
  764. Four Incident Circles [Java]
  765. Four Incircles in Equilateral Triangle [Java]
  766. Four Isosceles Triangles Hinged at Vertices [JavaScript, GeoGebra]
  767. Four 9-Point Circles in a Quadrilateral [Java]
  768. Four Pedal Circles [Java]
  769. Four Touching Circles [Java]
  770. Four Touching Circles II [Java]
  771. Four Touching Circles III [Java]
  772. The Four Touching Circles Problem IV [Java, GeoGebra]
  773. Four Triangles, One Circle [Java]
  774. Four Turtles (or Bugs, or Dogs, etc.) [Java]
  775. Fractal curves and dimension [Java]
  776. François Viète's Reduction of CCC to PCC [Java, GeoGebra]
  777. Friendly Kiepert's Perspectors [Java]
  778. From 3 to Golden Ratio in Semicircle
  779. From Angle Bisector to 120°120 degrees Angle
  780. From Arbitrary Hexagon to Regular One [JavaScript, GeoGebra]
  781. From Arbitrary Pentagon to Regular One [JavaScript, GeoGebra]
  782. From Equilateral Triangle and Square to Golden Ratio
  783. From Foci to a Tangent in Ellipse [Java]
  784. From One Collinearity to Another
  785. From Perpendicular Center Lines to Concyclic Points
  786. From Straight Line to Regular Hexagon [JavaScript, GeoGebra]
  787. From Triangle Inequality to Inequality in Triangle
  788. Fun Identity in Right Triangle
  789. Functions And Triangles
  790. Fundamental Theorem of 3-Bar Motion [Java]
  791. Further Arithmetic Progression in 30-60-90 Triangle
  792. Further Properties of Peculiar Circles
  793. Further properties of Van Aubel Configuration [JavaScript, GeoGebra]
  794. Fuss' Theorem [Java]
  795. Garcia's Archimedean Quadruplets [JavaScript, GeoGebra]
  796. Garcia's Two Circles Lemmas [JavaScript, GeoGebra]
  797. Garcia-Feuerbach Collinearity [JvaScript, GeoGebra]
  798. Garfield's Ghost Revisited - by Tony Foster
  799. Generalizing a Romanian Olympiad Problem [Java]
  800. Geometric/Trigonometric Observation/Exercise
  801. Geometric Illustration of a Convergent Series
  802. Geometric Mean In Trapezoid
  803. Geometric Mean Sangaku
  804. Geometric Meaning of the Geometric Mean
  805. Geometric Optimization from the Asian Pacific Mathematical Olympiad [Java]
  806. Geometric Proof of Hlawka's Inequality
  807. Geometric Proofs Of the Irrationality of Square Roots
  808. Geometry, Algebra, and Illustrations
  809. Gergonne and Medial Triangles Are Orthologic [Java]
  810. Gergonne and Soddy Lines Are Perpendicular [Java]
  811. Gergonne in Ellipse [Java]
  812. Gergonne and Van Obel Theorems
  813. Getting the first digits of Pi by counting collisions
  814. Gion Shrine Problem
  815. Given Parabola, Find Axis
  816. Glide Reflection [Java]
  817. Golden Ratio in Geometry
  818. Golomb's inductive proof of a tromino theorem [Java]
  819. Golden Ratio in Wu Xing
  820. Golden Ratio In Three Circles And Common Secant
  821. Gothic Arc [Java]
  822. Graph Theory [Java]
  823. Grebe: from Ladies' Diary to Carroll's Pillow [Java]
  824. Grebe's Theorem
  825. Greg Markowsky's Problem for Parabola [Java]
  826. Gregoie Nicollier's Proof of Napoleon's Theorem
  827. Griffiths Points in Cyclic Quadrilateral [Java]
  828. Griffiths' Theorem
  829. Half Turn, Reflection in Point [Java]
  830. Halving a square
  831. Harmonic Mean in Geometry
  832. Harmonic Mean Sangaku [Java]
  833. Harmonic Ratio
  834. Harmonic Ratio in Complex Domain
  835. Hart Circle [Java]
  836. Hart's Inversor [Java]
  837. Haruki's Lemma [Java]
  838. Haruki's Theorem
  839. Hausdorff Distance
  840. Hearty Munching on Cardioids [Java]
  841. Height/Width Interplay Illusion [Java]
  842. Helly's Theorem
  843. Hering Illusion [Java]
  844. Hermann Grid Illusion [Java]
  845. Heron's formula
  846. Heron's Formula: a Proof [Java]
  847. Heron's Problem [Java]
  848. Heron in Trapezoid [JavaScript, GeoGebra]
  849. Hexagon Parallel to Medial Triangle [Java]
  850. Hexagon Parallel to Orthic Triangle [Java]
  851. Hexagons in a Square
  852. Hidden Orthogonality [JavaScript, GeoGebra]
  853. Hinged Greek Cross Tessellation [Java]
  854. Hinged Squares [Java]
  855. Hinged Squares II [JavaScript, GeoGebra]
  856. Hippocrates' Squaring of a Lune [Java]
  857. Hippocrates' Squaring of Lunes
  858. Hjelmslev Theorem [Java]
  859. Hlawka in Convex Quadrilateral
  860. Homologous Lines under Three Spiral Similarities [Java]
  861. Homothety [Java]
  862. Homothety between In- and Excircles [Java]
  863. Homothety in Equilateral Triangles [JavaScript, GeoGebra]
  864. Homothety in Three Tangent Circles
  865. Homothety in Three Tangent Circles II
  866. Hooper's Paradox [Java]
  867. How Do Angle Trisectors Divide the Area?
  868. How Fast Does One's Shadow Grow?
  869. How Long Are the Angle Bisectors of a Degenerate Triangle?
  870. How to Construct a Radical Axis [Java, GeoGebra]
  871. How to Construct Tangents from a Point to a Circle [Java, GeoGebra]
  872. Hung Nguyen Viet's Inequality with Radicals and Chebyshev
  873. Hunting Right Angles [Java]
  874. Huygens' Problem
  875. Ian McGee's Observation [Java]
  876. IMO 1998 Problem 1
  877. IMO 2004, Problem 5
  878. IMO 2006, Problem 1
  879. Impossible Fork [Java]
  880. Impossible Frame [Java]
  881. Impossiblity Condition in Concurrent Cevians
  882. Importance of Having an Angle of 60 Degrees
  883. Importance of Occluders: Motion Binding [Java]
  884. In the Spirit of Thebault I [Java, GeoGebra]
  885. In the Wasan Spirit
  886. Incenters in Cyclic Quadrilateral [Java]
  887. Incidence in Feuerbach's Theorem [Java]
  888. Incircle in Equilateral Triangle
  889. Inequalities in Triangle
  890. Inequality in Quadrilateral
  891. Inequality in Rectangle
  892. An Inequality with a Variety of Circumradii
  893. Inequality with Cubes and Cube Roots
  894. An Inequality with Inradius and Excenters
  895. Inequality with Roots, Squares and the Area
  896. An Inequality with Two Sets of Cevians
  897. Infinitude of Primes - A Topological Proof
  898. Inradii Relation in Right Triangle
  899. Inscribed and Central Angles in a Circle [Java]
  900. Inscribed Angles [Java]
  901. Inscriptible and Exscriptible Quadrilaterals [Java]
  902. Inscriptible Quadrilateral: An Illustration [Java]
  903. Inscriptible Quadrilateral of Triangle Incenters [JavaScript, GeoGebra]
  904. Integer Triangles with Two Angles in the Ratio 1 : 2 [JavaScript, GeoGebra]
  905. Internal Tangents to Three Circles [Java]
  906. Interpretation of the coordinate formula for the dot product in 3D
  907. Intersecting Chords Theorem
  908. Intersections of a Circle with the Four Quadrants [Java]
  909. Intouch Triangle in Poncelet Porism [Java]
  910. Invariance in Orthodiagonal Quadrilaterals [Java]
  911. Invariant Area Sweep Implies Pythagorean Theorem [Java]
  912. Inverse Similarity through Reflections
  913. Inversion: Reflection in a Circle [Java]
  914. Involution on the Projective Line
  915. Is Every Trapezoid Parallelogram? [Java]
  916. Is X a Midpoint of a Chord [JavaScript, GeoGebra]
  917. Isosceles Tetrahedra Characterization via Inequalities
  918. Isogonal Concurrencies [JavaScript, GeoGebra]
  919. Isogonal Lemma [JavaSCript, GeoGebra]
  920. Isosceles Configuration in Triangle [JavaScript, GeoGebra]
  921. Isosceles Tetrahedron
  922. Isosceles Trapezoid In Cyclic Quadrilateral [JavaScript, GeoGebra]
  923. Isosceles Triangle with Two Points on Base
  924. Isogonal and Isotomic Conjugalities [Java]
  925. Isogonal Concurrency [Java]
  926. Isogonal image of the circumcircle [Java]
  927. Isoperimetric Theorem and Inequality
  928. Isosceles on the Sides of a Triangle [Java]
  929. Isotomic Reciprocity [JavaScript, GeoGebra]
  930. Isotomy and Isogonality Hand-in-Hand [Java]
  931. Iterations and the Mandelbrot Set [Java]
  932. Iterations on a Circle Through Three Points [Java]
  933. Iterations on Euler Lines [Java]
  934. Igor Sharygin's Problem in Circumscribed Quadrilateral
  935. J. Casey's Additions to Euclid I.47
  936. Jan Stevens' proof of E. W. Dijkstra's Generalization of the Pythagorean theorem
  937. Joachimsthal's Notations
  938. John Sharp's Paradox [Java]
  939. Johnson Circles [Java]
  940. Join Circles by Given Segment [Java]
  941. Joined Common Chords of Napoleon's Circumcircles [JavaScript, GeoGebra]
  942. Jordan and Kober Inequalities, PWW
  943. Jordan Curve Theorem
  944. Joseph Keech in Bride's Chair [Java]
  945. Judd Illusion [Java]
  946. Julia sets indexing [Java]
  947. K. Knop's Problem with Two Regular Pentagons And an Equilateral Triangle
  948. Kanizsa Triangle [Java]
  949. Kiepert's Centroid [JavaScript, GeoGebra]
  950. Kiepert's Theorem [Java]
  951. Kiepert's Triangles Graduate to Ears of Arbitrary Shape [JavaScript, GeoGebra]
  952. Klamkin's Quickie via Peru Geometrico
  953. Knots [avi movie]
  954. Knots on a Torus [Java]
  955. Kürschak's Tile and Theorem [Java]
  956. La Hire's Theorem [Java]
  957. Langman's Paradox [Java]
  958. LCC: Apollonius' Problem with Two Circles and a Line [Java, GeoGebra]
  959. Lean Napoleon's Triangles [Java]
  960. Length of a Median
  961. Length of the diagonal of the unit square equals the square root of 2
  962. Lennes' Polyhedron [Java]
  963. Leo's Lemma, Second Application
  964. Leo Giugiuc's Inequality for the Medians
  965. Leo Giugiuc's Inequality in Triangle, Solely with Cotangents
  966. Leo Giugiuc's Proof of Napoleon's Theorem
  967. Leo Giugiuc's Trigonometric Lemma
  968. Leo Giugiuc's Second Lemma And Applications
  969. Leonardo's Petals
  970. Lepidoptera of the Circles [Java]
  971. Leuenberger's Inequality for Medians, Inradius and Circumradius
  972. Lighthouse at Fermat Points [JavaScript, GeoGebra]
  973. Line IO in Bicentric Quadrilaterals
  974. Line Not Through a Center of Similarity [JavaScript, GeoGebra]
  975. Line Through a Center of Similarity [JavaScript, GeoGebra]
  976. Line through a Point in an Angle
  977. Line, Circle, and Fixed Points [Java]
  978. Linear Function with Coefficients in Arithmetic Progression [Java, GeoGebra]
  979. Lines Crossing Circles at Vertices of Equilateral Triangle
  980. Lines through Circles at Vertices of Equilateral Triangle [JavaScript]
  981. LLC: Apollonius' Problem with Two Lines and a Circle [Java, GeoGebra]
  982. Lipogrammatic Proof of the Pythagorean Theorem
  983. Loci of points in a cube
  984. Locus of Points in a Given Ratio to Two Points [Java]
  985. Longest segment
  986. Longitude, Latitude and Distance to the Equator
  987. Looking Back at Bottema [JavaScript, GeoGebra]
  988. Mach's Conservation of Area and the Pythagorean Theorem
  989. MacLaurin's Construction of Conics [Java]
  990. Maimed Cake [Java]
  991. Malfatti's Problem [Java]
  992. Malfatti's Problem, Hart's Solution [Java]
  993. Marian Cucoanes' Inequality With Roots and Powers
  994. Marian Dinca's Inequality
  995. Mascheroni Construction of a Regular Pentagon
  996. Mathematical Droodles [Java]
  997. Mathematics in Pizzeria [Java]
  998. MathPro Logo [Java]
  999. Maximal Properties of the Pythagorean Relation
  1000. Maximum Area Property of Equilateral Triangles
  1001. Maximum Area Property of the Medial Triangle
  1002. Maximum Perimeter Property of the Incircle [Java]
  1003. Maximum Ratio of Similarity - Belarus Olympiad Problem
  1004. Maxwell's Theorem [Java]
  1005. May the Product of Planes Be a Sphere?
  1006. McDougall's Generalization of Ptolemy's Theorem
  1007. Mechanical Proofs of the Pythagorean Theorem
  1008. Median in Touching Circles [Java]
  1009. Medians
  1010. Medians in a Quadrilateral [Java]
  1011. Medians in a Triangle Meet at the Center: a Second Look
  1012. Meisters' Two Ears Theorem
  1013. Menelaus Theorem [Java]
  1014. Metric Relations in a Triangle
  1015. Mickey Might Be a Red Herring in the Mickey Mouse Theorem
  1016. Midline in Quadrilateral [Java, GeoGebra]
  1017. Midline in Similar Triangles [Java]
  1018. Midline in Trapezoid [Java, GeoGebra]
  1019. Midline in Triangle [Java, GeoGebra]
  1020. Midpoint Reciprocity in Napoleon's Configuration [JavaScript,GeoGebra]
  1021. Midpoints and Orthogonality in Isosceles Triangles [Java]
  1022. Midpoints from Gergonne Triangle [Java]
  1023. Midpoints of the Lines Joining In- and Excenters [Java]
  1024. Miguel's Area of Square [JavaScript, GeoGebra]
  1025. Miguel's Tangents: Modern Day Sangaku [JavaScript, GeoGebra]
  1026. Miguel Ochoa's Chords And Tangents [JavScript, GeoGebra]
  1027. Miguel Ochoa's van Schooten Is a Slanted Viviani [JavScript, GeoGebra]
  1028. Miguel Ochoa's van Schooten Like Theorem [JavaScript, GeoGebra]
  1029. Miguel Ochoa's van Schooten Like Theorem II [JavaScript, GeoGebra]
  1030. Miguel Ochoa's van Schooten Like Theorem III [JavaScript, GeoGebra]
  1031. Miguel's Special Trapezoid
  1032. Mikhalev's Octahedrons
  1033. Minesweeper Theorem
  1034. Miquel's Point [Java]
  1035. Miquel's Point of a 4-line Via Spiral Similarity [Java]
  1036. Miquel's Theorem for Circles [Java]
  1037. Miquel Circumcenters [JavaScrirpt, GeoGebra]
  1038. Mirror on the Wall [Java]
  1039. Mirror property of the altitudes [Java]
  1040. Mirror Property of Altitudes via Pascal's Hexagram [Java]
  1041. Mixing Incenter And Orthocenter in Greece
  1042. Mixtilinear Circles and Concurrence [Java]
  1043. Modern Day Sangaku
  1044. Moebius Strip [avi movie]
  1045. Möbius' Strip [Java]
  1046. Models and Metamathematics
  1047. Moment of Inertia of a Regular Polygon [JavaScript]
  1048. Moments in a Peculiar Cyclic Polygon
  1049. Monge via Desargues [Java]
  1050. Monge via Desargues II [Java]
  1051. More of More of Gloden Ratio in Equilateral Triangles
  1052. Morley Constellation [Java]
  1053. Morley's Miracle [Java]
  1054. Morphing [Java]
  1055. Müller-Lyer Illusion [Java]
  1056. Multifaceted Cork [Java]
  1057. Multiplication of Points on a Circle [JavaScript,GeoGebra]
  1058. Multiplication of Points on an Ellipse [JavaScript, GeoGebra]
  1059. Munching on Circles
  1060. Munching on Inscribed Angles
  1061. My logo
  1062. Nagel Line [Java]
  1063. Nagel point [Java]
  1064. Nagel Point and Centroid on Adjacent Isosceles Triangles
  1065. Nagel Point of the Medial Triangle [Java]
  1066. Nagel Point on the Incircle in Isosceles Triangle
  1067. Nagel's Theorem [Java]
  1068. Napoleon's Hexagon [JavaScript, GeoGebra]
  1069. Napoleon's Theorem
  1070. Necker Cube [Java]
  1071. Neuberg Cubic [Java, GeoGebra]
  1072. Neuberg Sangaku [Java]
  1073. Newly Born Pair of Siblings to Archimedes' Twins [Java]
  1074. Newton's Construction of Conics [Java]
  1075. Newton's Theorem [Java]
  1076. Nine Point Center in Square
  1077. Nine Point Circle [Java]
  1078. Nine Point Circle: an Elementary Proof [Java]
  1079. No Equilateral Triangles, Please [Java]
  1080. Nobbs' Points, Gergonne Line [Java]
  1081. Non-Eclidean Geometries
  1082. No-Pedal Collinearity [Java]
  1083. Not So Hidden Homotheties [JavaScript, GeoGebra]
  1084. Novel Concurrency of Perpendicular Bisectors [JavaScript, GeoGebra]
  1085. Number of Regions N Lines Divide Plane
  1086. Occluded Wheel Illusion [Java]
  1087. Olof Hanner's Jigsaw Puzzle [Java]
  1088. On the Difference of Areas
  1089. One More Property of Equilateral Triangles
  1090. One More Seven Circles Theorem [JavaScript, GeoGebra]
  1091. One Sheet Hyperboloid [Java]
  1092. A One-Sided Inequality in Triangle
  1093. One Triangle from Another
  1094. One Triangle from Another II
  1095. Optimal Quadrilateral Inscribed in a Square
  1096. Optimization in a Crooked Trapezoid
  1097. Optimization in Parallelepiped
  1098. Optimization Problem in Acute Angle [Java]
  1099. Octagon in Parallelogram: Overlapping Regions [Java]
  1100. Octagon in Parallelogram [Java]
  1101. Optical Property of Ellipse [Java]
  1102. Orbison illusion [Java]
  1103. Orthic Semiperimeter [Java, GeoGebra]
  1104. Orthic Triangle in a Triangle with a 120 degrees angle
  1105. Orthogonal Lines, Midpoints, and Collinearity [Java, GeoGebra]
  1106. Orthogonality in Isogonal Conjugacy [JavaScript, GeoGebra]
  1107. Orthocenter and Three Equal Circles [Java]
  1108. Orthocenters
  1109. Orthocenters of Two Triangles Sharing Circumcenter and Base [Java]
  1110. Orthocentric System From Rectangle [Java]
  1111. Orthodiagonal and Cyclic Quadrilaterals [Java]
  1112. Orthogonality in Isosceles Triangles [Java]
  1113. Orthogonality in Two Squares [JavaScript, GeoGebra]
  1114. Orthologic Triangles in a Quadrilateral [Java]
  1115. Orthopole [Java]
  1116. Out of Pentagon Sangaku
  1117. Packing shapes in a plane
  1118. Padoa's Inequality
  1119. Pairs of Areas in Equilateral Triangle
  1120. Pairs of Homologous Lines under Spiral Similarities [Java]
  1121. Pairs of Incircles in a Quadrilateral [Java]
  1122. Pantograph [Java]
  1123. Paper Folding & Cutting Sangaku
  1124. Paper Folding Geometry
  1125. Paper strip activities [avi movies]
  1126. Paper strip activities, two strips
  1127. Pappus' Generalization of Euclid I.47 [Java]
  1128. Pappus' Theorem [Java]
  1129. Parabola [Java]
  1130. Parabola and Abutting Equilateral Triangles
  1131. Parabolas Related to the Orthic Triangle [JavaScript, GeoGebra]
  1132. Parabolic Mirror, Illustration [Java]
  1133. Parabolic Mirror, Theory [Java]
  1134. Parabolic Reciprocity [JavaScript, GeoGebra]
  1135. Parabolic Sieve of Prime Numbers
  1136. Paragon [Java]
  1137. Parahexagon [Java]
  1138. Parallel Chords [Java]
  1139. Parallel Chords II [Java]
  1140. Parallel Chords in Conics [Java, GeoGebra]
  1141. Parallel Chords in Ellipse [Java]
  1142. Parallel Chords in Crossing Circles [Java]
  1143. Parallel Lines in a Cyclic Quadrilateral [Java]
  1144. Parallel Lines in a Quadrilateral II [Java] Parallel Lines in a Quadrilateral
  1145. Parallel Lines in Rhombus
  1146. Parallel Lines in a Triangle [Java, GeoGebra]
  1147. Parallels through the Vertices of Equilateral Triangle
  1148. Parallelogram and Ellipses [Java, GeoGebra]
  1149. Parallelogram and Four Equilateral Triangles [JavaScript, GeoGebra]
  1150. Parallelogram and Similar Triangles [Java]
  1151. Parallelogram Illusion [Java]
  1152. Parallelogram in Four Circles [JavaScript, GeoGebra]
  1153. Parallelogram in Parallelogram And in Projection [JavaScript, GeoGebra]
  1154. Parallelogram in Trapezoid
  1155. Parallelogram Iterations [Java]
  1156. Parallelogram Law [Java]
  1157. Parallelogram Law: A PWW [Java]
  1158. Parallelogram Law: A Tessellation [Java]
  1159. Parallelogram with Side Lines through Fixed Points [JavaScript, GeoGebra]
  1160. Partition of point sets in a plane
  1161. Partitioning 3-Space with Circles
  1162. Partitioning a Circle
  1163. Pascal: Necessary and Sufficient [Java]
  1164. Pascal in a Cyclic Quadrilateral [Java]
  1165. Pascal Lines: Steiner and Kirkman Theorems [Java]
  1166. Pascal Lines: Steiner and Kirkman Theorems II [Java]
  1167. Pascal's theorem [Java]
  1168. Peacock's Tail Sangaku
  1169. Peaucellier Linkage [Java]
  1170. Pedal Collinearities [Java]
  1171. Pedal Parallelogram [JavScript, GeoGebra]
  1172. Pedal Polygons [Java]
  1173. Pedal Property in Equilateral Triangle
  1174. Pedal SimilarityPedal Triangle and Isogonal Conjugacy [Java]
  1175. Pedal Triangles of Inverse Points [JavaScript, GeoGebra]
  1176. Pedoe's Theorem [Java]
  1177. Pencils of Cubics
  1178. Pentagon, construction of
  1179. Pentagon And Decagon, Both Regular
  1180. Pentagon in a Semicircle [Java]
  1181. Pentagon Proportions Sangaku
  1182. Perimeters of Convex Polygons, One within the Other
  1183. Perimeters of Parallelogram And Rhombus [JavaScript, GeoGebra]
  1184. Periodic patterns in sin(N) [Java]
  1185. Perpendicular Bisectors in an Inscriptible Quadrilateral [Java]
  1186. Perpendicular Bisectors in an Inscriptible Quadrilateral II [Java]
  1187. Perpendicular lines in a cyclic quadrilateral from 1975 Canadian Mathematical Olympiad
  1188. Perpendiculars in Parallelogram [JavaScript, GeoGebra]
  1189. Perspective Triangles in Circle
  1190. Pick's Theorem [Java]
  1191. Phantom Circle and Recaptured Symmetry
  1192. Plane Filling Curves [Java]
  1193. Plane Isometries
  1194. Plane Isometries As Complex Functions
  1195. Plato's Geometric Numbers: can you better Sam Loyd?
  1196. Pleasant Proportions in Triangle
  1197. Poggendorff's Illusion
  1198. Point common to two similar rectangles [Java]
  1199. Point in a Square
  1200. Point on Bisector in Right Angle
  1201. Points Generated by the Nine Points
  1202. Points on Incircle: Another Look
  1203. Polar Circle [Java]
  1204. Poles and Polars [Java]
  1205. Polygon in a Rectangle
  1206. Polygons: formality and intuition [Java]
  1207. Polygons Formed by Perpendicular Bisectors [Java]
  1208. Pompeiu's Theorem
  1209. Poncelet's Porism [Java]
  1210. Poncelet's Porism in Ellipses [Java]
  1211. Poncelet's Theorem [Java]
  1212. Ponzo Illusion [Java]
  1213. Ponzo Illusion II [Java]
  1214. Possible or Impossible? [Java]
  1215. Power of a point with respect to a circle
  1216. PCC: Apollonius' Problem with Two Circles and a Point [Java, GeoGebra]
  1217. PLC: Apollonius Problem with a point, a line, and a circle [Java, GeoGebra]
  1218. PLL: Apollonius' Problem with Two Lines and a Point [Java, GeoGebra]
  1219. PPC: Apollonius' Problem with Two Points and a Circle [Java, GeoGebra]
  1220. PPL: An Elementary Solution [Java, GeoGebra]
  1221. PPL: Apollonius' Problem with Two Points and a Line [Java, GeoGebra]
  1222. Prasolov's Pythagorean Identity
  1223. Praying Eyes Theorem [JavaScript, GeoGebra]
  1224. Problem 1 from the EGMO2017
  1225. Problem 1 from the Ninth Nordic Mathematical Contest (1994)
  1226. Problem 11984 From the American Mathematical Monthly
  1227. Problem 3 from the EGMO2017
  1228. Problem 4 from the 1959 IMO: Construction of a Right Triangle [Java, GeoGebra]
  1229. Problem 4, 1975 USA Math Olympiad and the Radical Axis [Java]
  1230. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles [Java]
  1231. Problem 4, 1975 USA Math Olympiad: Normals and Tangents [Java]
  1232. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch [Java]
  1233. Problem 4 from the 2016 OBJM TST
  1234. Problem 4 from the IMO 2009
  1235. Problem 4 from the 4th BJMO Team Selection Test
  1236. Problem 4087 from Crux Mathematicorum
  1237. Problem 4096 from Crux Mathematicorum
  1238. Problem in an Equilateral Triangle [Java]
  1239. Problem 4 from the IMO 2013 - Synthetic Solution [Javacript, GeoGebra]
  1240. Problem CC156 From Crux Mathematicorum
  1241. Problem in a Special Trapezoid
  1242. Problem in an Equilateral Triangle II [Java]
  1243. Problem in Direct Similarity [JavaScript, GeoGebra]
  1244. Problem in Three Squares [Java]
  1245. Problem of Equal Steps [Java]
  1246. Problem of Equal Steps II [Java]
  1247. Problem on a Pentagonal Star
  1248. Problem with angle bisector from 1975 Canadian Mathematical Olympiad
  1249. Problem with equal angles from 1987 IberoAmerican Mathematical Olympiad
  1250. Proclivity of Conjoint Archimedean Twins for Proliferation [JabvaScript, GeoGebra]
  1251. Projections of Convex Quadrilateral [JavaScript, GeoGebra]
  1252. Projections of Isogonal Conjugate are Concyclic [JavaScript, GeoGebra]
  1253. Projections on Internal and External Angle Bisectors [Java]
  1254. Projective Collinearity in a Quadrilateral [Java]
  1255. Projective infinity
  1256. Projective Invariants for Pascal And Desargues
  1257. Properties of Flank Triangles
  1258. Properties of the Circle of Similitude [Java]
  1259. Properties of Circle Through the Incenter
  1260. Properties of the Figures in Euclid I.47
  1261. Properties of the Harmonic Ratio
  1262. Property of 6-Parpolygon [Java]
  1263. Property of Angle Bisectors [Java]
  1264. Property of Circumscribed Quadrilaterals [Java]
  1265. Property of Points Where In- and Excircles Touch a Triangle [Java]
  1266. Property of Right Isosceles Triangle [Java, GeoGebra]
  1267. Property of Right Trapezoids [GeoGrebra]
  1268. Property of Semicircles [Java]
  1269. Property of Thébault Circles [JavaScript, GeoGebra]
  1270. Property of Two Pencils of Parallel Lines
  1271. Proportions and the Incenter [JavaScript]
  1272. Proportions in Square
  1273. Proportions with Two Orthocenters [JavaScript, GeoGebra]
  1274. Ptolemy's Theorem
  1275. Pure Angle Chasing [JavaScript, GeoGebra]
  1276. Pure Angle Chasing II [JavaScript, GeoGebra]
  1277. Pure Angle Chasing III [JavScript, GeoGebra]
  1278. Puzzles on graphs.
  1279. Pythagorean Theorem (118 proofs)
  1280. Pythagorean Perimeter Theorem
  1281. Pythagoras and Vecten Break Japan's Isolation [Java]
  1282. Pythagorean Triples [Java]
  1283. Quadrature: A Child's Play
  1284. Quadrilateral Inequality
  1285. Quadrilateral from a Segment
  1286. Quadrilateral With Equal Opposite Sides And Angles [Java]
  1287. Quadrilateral with 3 Equal Sides [Java]
  1288. Quadrilaterals Formed by Perpendicular Bisectors [Java]
  1289. Quest for Paragon [Java]
  1290. R and r When G on Incircle
  1291. Radical Axis and Center, an Application [Java]
  1292. Radical Axis of Circles Inscribed in a Circular Segment [Java]
  1293. Radical Center [Java]
  1294. Radius and Construction of a Mixtilinear Circle [Java]
  1295. Radius of a Circle by Paper Folding [Java]
  1296. Random Dot Stereograms [Java]
  1297. The Range of r/R in Triangle with Centroid on the Incircle
  1298. Ratios and Orthogonality in a Triangle - Problem from the 2006 Romania MO
  1299. Rational Points On a Circle
  1300. Rectangle Inscribed in Triangle [JavaScript, GeoGebra]
  1301. Rectified, Halved, Sheared, Eyeballs Still Surprise [JavaScript, GeoGebra]
  1302. Reflection in a Triangle Tangent to Parabola [JavaScript, GeoGebra]
  1303. Reflection In Line [Java]
  1304. Reflection in the Circumcenter
  1305. Reflections in Ellipse [Java]
  1306. Reflections of a Line Through the Orthocenter [Java]
  1307. Reflections of a Point on the Circumcircle [Java]
  1308. Reflections of the Orthocenter
  1309. Reflections of the Orthocenter [Java]
  1310. Reflections of the Orthocenter II Regular Hexagon and Triangle Joined at Vertex [JavaScript, GeoGebra]
  1311. Regular Hexagon from Arbitrary Triangle [JavaScript, GeoGebra]
  1312. Regular Octagon by Paper Folding
  1313. Regular Polygons in a Triangular Grid [Java]
  1314. Regular Polyhedra - Platonic Solids
  1315. Other Solids
  1316. Regular Pentagon Inscribed in Circle by Paper Folding
  1317. Regular polyhedra and Euler's Theorem
  1318. Reim's Similar Coins I
  1319. Reim's Similar Coins II
  1320. Reim's Similar Coins III
  1321. Reim's Similar Coins IV
  1322. Relation Engendered By Altitudes
  1323. Relation Engendered By Medians
  1324. Relations in a Cyclic Polygon
  1325. Remarkable Line in a Quadrilateral [Java]
  1326. Remarkable Line in Cyclic Quadrilateral [Java]
  1327. Rendezvous [Java]
  1328. Reverse Spoke Illusion [Java]
  1329. Revolving Circles Illusion [Java]
  1330. Rhombus in a Cyclic Quadrilateral [Java]
  1331. Rhombus in Circles [Java, GeoGebra]
  1332. Riemann Sphere and Möbius Transformation
  1333. Surprise: Right Angle in Circle [JavaScript, GeoGebra]
  1334. Surprising Length Dependence In Equilateral Triangle by Miguel Ochoa Sanchez
  1335. Right Triangles on Sides of a Square [Java]

    Rotating Square in Search of the Golden Ratio

  1336. Rotating Top Illusion [Java]
  1337. Rotating Gears [Java]
  1338. Rotation Transform [Java]
  1339. Rouse Ball's Fallacy [Java]
  1340. Rusty Compass Construction of Equilateral Triangle [Java]
  1341. Salinon: From Archimedes'Book of Lemmas [Java]
  1342. Sam Loyd's Geometric Puzzle
  1343. Sam Loyd's Son's Dissection [Java]
  1344. Sanchez's Areas in Bottema's Configuration [JavaScript]
  1345. Sangaku: Reflections and the Phenomenon
  1346. Sangaku: Critique of My View and a Response
  1347. Sangaku and The Egyptian Triangle
  1348. Sangaku in a Square
  1349. Sangaku Iterations, Is it Wasan? [Java]
  1350. Sangaku via Peru
  1351. Sangaku with 8 Circles
  1352. Sangaku with Angle between a Tangent and a Chord [Java]
  1353. Sangaku with Quadratic Optimization
  1354. Sangaku with Three Mixtilinear Circles [Java]
  1355. Sangaku with Versines [Java]
  1356. Sangakus with a Mixtilinear Circle [Java]
  1357. Scintillating Grid Illusion [Java]
  1358. Schwarz Lantern [Java]
  1359. Secant Angles [Java]
  1360. Secant Angles II [Java]
  1361. Secant, Normal, Tangent [Java]
  1362. Secant, Tangents and Orthogonality [Java]
  1363. See-Saw Lemma [Java]
  1364. Segment Projections in Equilateral Triangle
  1365. Segment Trisection [Java]
  1366. Segment Trisection II
  1367. Segment Trisection Induced by Parallels to Medians [Java]
  1368. Semicircle on Square [JavaScript]
  1369. Semi-regular Tessellation on Hinges A [Java]
  1370. Semi-regular Tessellation on Hinges B [Java]
  1371. Separating point sets with a circle
  1372. Sequences of Concyclic Points on a Conic
  1373. Sequences of Touching Circles
  1374. Serendipitous Beauty [Java]
  1375. Serendipitous Proofs Of The Pythagorean Theorem
  1376. Seven and the Eighth Circle Theorem
  1377. Seven Circles Theorem [Java]
  1378. Seven Concyclic Points [Java, GeoGebra]
  1379. Seven Equal Circles [JavaScript, GeoGebra]
  1380. Seven Problems in Equilateral Triangle
    1. Seven Problems in Equilateral Triangle, Solution to Problem 1
    2. Seven Problems in Equilateral Triangle, Solution to Problem 2
    3. Seven Problems in Equilateral Triangle, Solution to Problem 3
    4. Seven Problems in Equilateral Triangle, Solution to Problem 4
    5. Seven Problems in Equilateral Triangle, Solution to Problems 5, 6, 7
  1381. Seyran Ibrahimov's Inequality
  1382. Shaggy Dog Theorem
  1383. Shapes in a lattice
  1384. Shapes of constant width [Java]
  1385. Shearing and Translation in Pythagorean Pants [Java]
  1386. Shearing a Polygon into a Triangle of Equal Area [Java]
  1387. Shearing Butterflies in Quadrilaterals [Java]
  1388. Shedding Light on the Ball for Eyeballing
  1389. Shepard's Parallelogram Illusion [Java]
  1390. Shifted by the Inradius Tangent Collinearities [JavaScript, GeoGebra]
  1391. Shifting Cube Illusion [Java]
  1392. Shimmer Illusion [Java]
  1393. Short Construction of the Geometric Mean
  1394. Shortest Fence in a Quarter-Circle Pasture
  1395. Shredding the torus
  1396. Sides and Area of Pedal Triangle
  1397. Sierpinski Gasket By Common Trema Removal [Java]
  1398. Sierpinski Gasket Via Chaos Game [Java]
  1399. Sierpinski Gasket by Trema Removal [Java]
  1400. Similar Parallelograms on Sides of a Triangle
  1401. Similar Triangles And More
  1402. Similar Triangles in Crossing Circles [JavaScript, GeoGebra]
  1403. Similar Triangles on Sides of a Quadrilateral [Java]
  1404. Similar Triangles on Sides and Diagonals of a Quadrilateral [Java]
  1405. Similarity Is About Proportion - And Ways to Look at It
  1406. Simple Construction of the Circle of Apollonius
  1407. Simple dissection of an acute triangle
  1408. Simple Property of Circle Through the Incenter [JavaScript]
  1409. Simple Quadrilaterals Tessellate the Plane [Java]
  1410. Simson Line [Java]
  1411. Simsons and 9-Point Circles in Cyclic Quadrilateral [Java]
  1412. Simultaneous Contrast Illusion [Java]
  1413. Simultaneous Contrast Illusion II [Java]
  1414. Simultaneous Contrast: Koffka's Illusion [Java]
  1415. Simultaneous Contrast: White's Illusion [Java]
  1416. Simultaneous Contrast: White's Illusion, V-Variant [Java]
  1417. Three Concurrent Chords at 60 Degrees Angles
  1418. Simultaneous Diameters in Concurrent Circles [Java]
  1419. Simultaneous Generalization of the Theorems of Ceva and Menelaus
  1420. Sine and Cosine of 15 Degrees Angle
  1421. Sine, Cosine, and Ptolemy's Theorem
  1422. Six Circles with Concurrent Pairwise Radical Axes
  1423. Six Circles Theorem (Bui Quang Tuan) [Java]
  1424. Six Circles Theorem (Elkies) [Java]
  1425. Six Circles Theorem (Evelyn) [Java]
  1426. Six Concyclic Circumcenters in Fermat's Configuration [JavaScript, GeoGebra]
  1427. Six Concyclic Points [Java]
  1428. Six Concyclic Points on Sides of a Triangle
  1429. Six Concyclic Points II [Java]
  1430. Six Concyclic Points via Antipedal Triangle [JavaScript, GeoGebra]
  1431. Six Incircles in an Equilateral Triangle [Java]
  1432. Six Point Circle [Java]
  1433. SixPoints, ThreeLines [Java]
  1434. Soddy Circles and David Eppstein's Centers [Java]
  1435. Some Properties of Common Tangents [JavaScript, GeoGebra]
  1436. Some Properties of Napoleon's Configuration [Java]
  1437. Spectral Decomposition of a Planar Quadrilateral
  1438. Sperner's Lemma [Java]
  1439. Spiral Similarity [Java]
  1440. Spiral Similarity and CollinearitySpiral Similarity and Collinearity [Java, GeoGebra]
  1441. Spiral Similarity Leads to Equilateral Triangle [JavaScript, GeoGebra]
  1442. Square and Circle in a Gothic Cupola
  1443. Square from Four Points, One on Each Side
  1444. Square From Nowhere [Java]
  1445. Square in a Circle Inscribed in a Square [Java]
  1446. Square in a Right Triangle [Java]
  1447. Square in Chair [Java]
  1448. Square Inscribed in a Triangle [Java]
  1449. Square Inscribed in a Triangle II [Java]
  1450. Square, Similarity and Slopes [Java]
  1451. Squares and Straight Tetrominoes
  1452. Squares in Circles [Java]
  1453. Squares in Semicircle and Circle
  1454. Squares in Triangles
  1455. Squares Inscribed In a Triangle III [Java]
  1456. Squares on Sides of a Parallelogram [JavaScript, GeoGebra]
  1457. Squares on Sides of a Quadrilateral [Java]
  1458. Squares on Sides of a Quadrilateral II [JavaScript, GeoGebra]
  1459. Square out of a Quadrilateral
  1460. Squaring a circle
  1461. Squaring a Rectangle [Java]
  1462. Squinting Eyes Theorem [Java]
  1463. SSA
  1464. SSS
  1465. Staircase Illusion [Java]
  1466. Stan Fulger's Observation in Right Triangle
  1467. Star of David [Java]
  1468. Stathis Koutras' Butterfly
  1469. Stathis Koutras' Theorem
  1470. Stathis Koutras' Theorem II
  1471. Steiner's Circumellipse in Barycentric Coordinates [GeoGebra]
  1472. Steiner's Inellipse
  1473. Steiner's Sangaku
  1474. Steiner's Ratio Theorem
  1475. Steiner's Theorem
  1476. Stereographic Projection [Java]
  1477. Stereographic Projection and Inversion
  1478. Stereographic Projection and Radical Axes
  1479. Stewart's theorem
  1480. Straight Edge Only Construction of Polar
  1481. Structural Constellation [Java]
  1482. Sums of Squares in Circle
  1483. Sum of Squares in Equilateral Triangle
  1484. Sum of Squares of Distances to Vertices
  1485. Sum of Two Squares [JavaScript, GeoGebra]
  1486. Sums of Unit Vectors
  1487. Surprising 60° in Equilateral Triangle [JavaScript, GeoGebra]
  1488. Sylvester's Problem
  1489. Symmedians
  1490. Symmetries in a Triangle
  1491. Symmetries in Triangle II [JavaScript, GeoGebra]
  1492. Symmetry in an Asymmetric Configuration [JavaScript, GeoGebra]
  1493. (tan(72°)±tan( 60°)sin(30°$\mp$18°)= 1
  1494. Tangency Point of Two Circles [Java]
  1495. Tangent and Secant [Java]
  1496. Tangent as a Radical Axis [Java]
  1497. Tangent Circles, Similarity and Equal Segments [JavaScript, GeoGebra]
  1498. Tangent Circles and an Isosceles Triangle [Java]
  1499. Tangent Circles in a Parallelogram [Java]
  1500. Tangent, Cotangent and Square Roots Inequality
  1501. Tangent Curiosity in Equilateral Triangle [JavaScript, GeoGebra]
  1502. Tangent in Concurrency [Java]
  1503. Tangent Lines and Circles in Convex Quadrilateral [Java]
  1504. Tangent of 22.5° - Proof Wthout Words
  1505. Tangent to Circle in Three Steps
  1506. Tangent to the Incircle [JavaScript]
  1507. Tangent to two circles
  1508. Tangents And Bisectors
  1509. Tangents and Diagonals in Cyclic Quadrilateral [Java]
  1510. Tangents, Perpendiculars and Geometric Mean [Java]
  1511. Tarski-Banach Decompositions
  1512. Taylor circle [Java]
  1513. Terquem's Theorem [Java]
  1514. Tesseract [Java]
  1515. Tetrahedron in Parallelepiped
  1516. Tetrahedron with Equiareal Faces
  1517. Tetrahedron with Equiareal Faces, II
  1518. Tetrahedron with Orthogonal Perpendiculars to Opposite Edges
  1519. Thales' Theorem [Java, GeoGebra]
  1520. Thales' Theorems
  1521. Thales on Angle Bisectors [JavaScript, GeoGebra]
  1522. The 80-80-20 Triangle
  1523. The Beauty of Fractions
  1524. The Center of Gravity and Motion of Material Points [JavaScript, GeoGebra]
  1525. The Chaos Game: Address Space vs IFS [Java]
  1526. The Devil is in the Exceptions
  1527. The Eutrigon Theorem [Java, GeoGebra]
  1528. The Golden Pentacross
  1529. The Law of Cosines by Larry Hoehn
  1530. The Midpoint of an Angle Bisector - a Problem by A. A. Zaslavsky
  1531. The only rational triangle
  1532. The Pythagorean Theorem from a Combinatorial Identity
  1533. The Schwarz Lantern Explained [JavaScript]
  1534. The Shape of the Colosseum [Java, GeoGebra]
  1535. The Shortest Crease [JavaScript, GeeoGebra]
  1536. Thébault's Problem I [Java]
  1537. Thébault's Problem II [Java]
  1538. Thébault's Problem III [Java]
  1539. Theorem of Complete Quadrilateral [Java]
  1540. Theorem of Three Tangents to a Conic [JavaScript, GeoGebra]
  1541. Theorems of Ceva and Menelaus, an Illustrated Generalization [Java]
  1542. There is no Difference Between Equilateral Triangles
  1543. Third Unusual Identity in Triangle
  1544. Three Circles and Area [JavaScript]
  1545. Three Circles and Common Tangents
  1546. Three Circles through the Incenter
  1547. Three Collinear and Four Concyclic Points [Java, GeoGebra]
  1548. Three Concurrent Circles [Java]
  1549. Three Common Chords in Three Concurrent Circles [Java, GeoGebra]
  1550. Three Congruent Circles by Reflection [Java]
  1551. Three Congruent Circles by Reflection II [Java]
  1552. Three Congruent Circles by Reflection III [Java]
  1553. Three Equal Circles in a Semicircle
  1554. Three Euler Lines That Are Four [JavaScript]
  1555. Three Incircles In a Right Triangle [Java]
  1556. Three Incircles in a Triangle [Java]
  1557. Three Isosceles Triangles [Java]
  1558. Three Orthogonal Circles Through Three Given Points [Java]
  1559. Three lines at 60 degres and Circumcenters [JavaScript, GeoGebra]
  1560. Three pairs of Parallel Lines and a Conic [JavaScript, GeoGebra]
  1561. Three Parabolas with Common Directrix [JavaScript,GeoGebra]
  1562. Three Parallels in a Triangle [Java]
  1563. Three Points on a Parabola [Java]
  1564. Three Pyramids are Better Than Two [Java]
  1565. Three regions sharing a common boundary
  1566. Three Similar Polygons [Java]
  1567. Three Similar Triangles [Java]
  1568. Three Similar Triangles II [Java]
  1569. Three Squares and Two Ellipses [Java]
  1570. Three Tangent Circles [Java]
  1571. Three Tangents, Three Chords in Ellipse [Java]
  1572. Three Tangent Circles Sangaku
  1573. Three Tangents Theorem [Java]
  1574. Three Tangents, Three Secants [Java]
  1575. Three Touching Circles [Java]
  1576. Three Travelers in Uniform Motion
  1577. Three visions of a fact [GeoGebra]
  1578. Tiling a Chessboard with Dominoes
  1579. Tiling Sphere by Four Equal Triangles
  1580. Toothpick Construction of a Square [Java]
  1581. Topological preliminaries
  1582. Torque as a Mathematical Object
  1583. Touching Circles [Java]
  1584. Touching Circles and Concurrency [Java]
  1585. Touching Circles with Given Centers II [Java]
  1586. Train wheel quandary
  1587. Transitivity in Action [Java]
  1588. Translated Triangles [Java]
  1589. Translation Transform [Java]
  1590. Transversal Trigonometry [JavaScript, GeoGebra]
  1591. Triangle ABC is right iff sin²A + sin²B + sin²C = 2
  1592. Triangle Area in Square with Incircle
  1593. Triangle Areas in a Parallelogram [Java]
  1594. Triangle Areas in a Parallelogram II [Java]
  1595. Triangle by HM segments [JavaScript, GeoGebra]
  1596. Triangle Classification [Java]
  1597. Triangle constructions and facts [Java]
  1598. Triangle Construction from Angle, Altitude and Median
  1599. Triangle from a, A, mc/mb
  1600. Triangle from Angle Bisector, Altitude and Circumradius
  1601. Triangle from Angle Bisector, Altitude and Inradius
  1602. Triangle from Inradius, Circumradius, Side or Angle
  1603. Triangle from Median, Altitude and Circumradius
  1604. Triangle from Side, Circumradius, and Median
  1605. Triangle from Angle, Inradius, and Difference of Sides
  1606. Triangle from Side, Inradius, and Altitude
  1607. Triangle from Side, Median, and Altitude
  1608. Triangle from Two Side Lines and the Euler Line
  1609. Triangle in a 4x4 Square
  1610. Triangle in a 5x5 Square
  1611. Triangle of Maximum Area
  1612. Triangle of Maximum Area II
  1613. Triangle of Minimum Perimeter [JavaScript]
  1614. Triangle Sliding on Two Straight Lines [Java]
  1615. Triangles, Squares and Areas from Temple Geometry [Java]
  1616. Triangles on HO
  1617. Triangles with Equal Area [Java]
  1618. Triangles with Sides in Geometric Progression [JavaScript]
  1619. Triangular Areas in Circle [JavaScript, GeoGebra]
  1620. Triangular Billiard [Java]
  1621. Triangulating Squares
  1622. Triangle with a 60 degrees angle [JavaScript, GeoGebra]
  1623. Trigonometry by Watching
  1624. Trigonometric Form of Ceva's Theorem [Java]
  1625. Trigonometric Functions [Java]
  1626. Trigonometric Identities with Arctangents
  1627. Trigonometric Proof of the Pythagorean Theorem
  1628. Trigonometry According to Ptolemy
  1629. Trigonometry by Paper Folding
  1630. Trisect Segment: 2 Circles, 4 Lines
  1631. Trisectable angle that is not constructible
  1632. Tucker Circles [Java]
  1633. Tucker Circles Through Homothety [Java]
  1634. Twelve Matchsticks Area Puzzle
  1635. Twin Segments in Four Semicircles
  1636. Twins Beget Twins
  1637. Twisted Cord Illusion [Java]
  1638. Two Altitudes, One Midpoint
  1639. Two Arbelos, Two Chains
  1640. Two Cevians and Proportions in a Triangle
  1641. Two Circles, Two Segments - One Ratio
  1642. Two Circles and a Limit - Analytic Proof
  1643. Two Circles and One More [Java]
  1644. Two Circles and Two More [Java]
  1645. Two Circles, Ellipse, and Parallel Lines
  1646. Two Circles in a Parallelogram [Java]
  1647. Two Circles Inscribed in a Parallelogram [Java]
  1648. Two Circles in a Square [Java]
  1649. Two Circles in a Square II [Java]
  1650. Two Circles in an Angle [Java]
  1651. Two Circles on Angle Bisector [Java]
  1652. Two Circles on a Side of a Triangle [Java]
  1653. Two Circles, Tangents and Two Collinearities [JavaScript, GeoGebra]
  1654. Two Circumcircles And Two Pairs of Parallels [Java]
  1655. Two Circumcircles in Triangle [Java]
  1656. Two Conditions for a Triangle to Be Equilateral
  1657. Two coins puzzle
  1658. Two Congruent Circles by Reflection [Java]
  1659. Two Conics, Pascal, Chasles, and Cross-Ratio [JavaScript, GeoGebra]
  1660. Two Color Coloring of the Plane [Java]
  1661. Two Common Chords in Three Concurrent Circles [Java, GeoGebra]
  1662. Two Diameters and Longest Common Chord [Java, GeoGebra]
  1663. Two Equilateral Triangles [Java]
  1664. Two Homotheties in a Parallelogram [Java]
  1665. Two Intersecting Circles [Java]
  1666. Two Lines - Two Circles [Java]
  1667. Two Pencils of Parallel Lines [JavaScript, GeoGebra]
  1668. Two Pairs of Parallel Lines in a Triangle [Java, GeoGebra]
  1669. Two Parallels in a Triangle and One More [Java]
  1670. Two Pascals Merge into One [JavaScript, GeoGebra]
  1671. Two Perpendiculars From a Point to a Line [Java]
  1672. Two Planar Constructions Related to a 3D Problem
  1673. Two Perspective Triangles
  1674. Two Quadrilaterals [Java]
  1675. Two Quadruplets of Concyclic Points [JavaSCript,GeoGebra]
  1676. Two Quadruplets of Tangent Circles [Java]
  1677. Two Refinements of Ionescu-Weitzenbock Inequality
  1678. Two Regular Pentagons Joined at Vertex
  1679. Two Related Triangles of Equal Areas
  1680. Two Sangaku with Equal Incircles
  1681. Two Similar Triangles with Parallel Sides [JavaScript, GeoGebra]
  1682. Two Simsons in a Triangle [Java]
  1683. Two Squares and Another Square [Java]
  1684. Two Squares In Circle [JavaScript]
  1685. Two Symmetric Triangles Are Directly Decomposable [Java]
  1686. Two Touching Circles [Java]
  1687. Two Triangles Inscribed in a Conic [Java, GeoGebra]
  1688. Two Triangles Inscribed in a Conic - with Elementary Solution [Java, GeoGebra]
  1689. Two Triangles Inscribed in a Conic - with Solution [Java, GeoGebra]
  1690. Two Triangles With Common Base and Altitude [Java]
  1691. Two Triangles Sharing the Circumcenter
  1692. Two Triples of Concurrent Cevians and a Conic [JavaScript, GeoGebra]
  1693. Two Triples of Concurrent Circles [Java]
  1694. Two Triples of Similar Triangles [Java]
  1695. Uncommon Triangle Constructions
  1696. Unfolding Spirals Illusion [Java]
  1697. Urquhart's Theorem
  1698. USA 1975 Problem 4
  1699. Use of Homogeneous Coordinates for Geometrical Calculations
  1700. Van Khea's Fourth Identity in Triangle
  1701. Van Khea's Quickie
  1702. Van Obel Theorem, Determinants, and Barycentric Coordinates
  1703. Van Schooten's and Pompeiu's Theorems [Java]
  1704. Van Schooten's Locus Problem [Java]
  1705. Variations on the Theme of Tremas [Java]
  1706. Variations on the Theme of Tremas II [Java]
  1707. Varignon Parallelogram [Java]
  1708. Various Geometries
  1709. Vecten-Nikolin Hexagon [Java]
  1710. Vecten's Collinearity [Java]
  1711. Vecten's Mesh [Java]
  1712. Vector Addition and Subtraction [Flash]
  1713. Vector Algebra in Tetrahedron
  1714. Vector product
  1715. Versatile Theorem [Java]
  1716. Vertical and Horizontal Dominoes on a Chessboard
  1717. Vietnamese Extension of a Japanese Theorem [JavaScript]
  1718. Vietnamese Lemma towards an Extension of a Japanese Theorem [JavaScript, GeoGebra]
  1719. Viewing a Statue
  1720. Viviani's Theorem [Java]
  1721. Viviani's Theorem II [Java]
  1722. Viviani in Isosceles Triangle [Java]
  1723. Viviani by Inversion
  1724. Viviani by Vectors
  1725. Viviani's 3D Analogue
  1726. Sanchez's Viviani's Area Analogue [JavaScript, GeoGebra]
  1727. Volume and Area of Torricelli's Trumpet [JavaScript]
  1728. Volume of Fibonacci Tetrahedron
  1729. Volume of Sphere and Volume of an Ellipsoid
  1730. Volumes of Two Pyramids
  1731. Wallace-Bolyai-Gerwien Theorem
  1732. Wallace's Theorem [Java]
  1733. Watt's and Chebyshev's Linkage [Java]
  1734. Weird curves bound normal areas
  1735. Weitzenbock by Sanchez
  1736. Weitzenböck Inequality [Java]
  1737. What does one see inside a spherical mirror? [Java]
  1738. What Is Analytic Geometry?
  1739. What Is Angle?
  1740. What Is Angle Chasing?
  1741. What Is Area?
  1742. What Is Ellipse?
  1743. What Is Indirect Proof?
  1744. What Is Line?
  1745. What Is Point?
  1746. When a Triangle is A-cute
  1747. When a Triangle is Equilateral [Java]
  1748. When Is Triangle Equilateral: Marian Dinca's Criterion
  1749. When Is Triangle Isosceles: Miguel Ochoa Sanchez's Criterion
  1750. When Two Segments Intersect? [JavaScript]
  1751. When index equals content [Java]
  1752. Wonderful Trigonometry In Equilateral Triangle
  1753. Why a Diameter Is the Longest Chord?
  1754. William Wallace's Proof of the Butterfly Theorem
  1755. William Wallace's 1803 Statement of the Butterfly Theorem [JavaScript, GeoGebra]
  1756. Wittenbauer's Parallelogram [Java]
  1757. Wundt Illusion [Java]
  1758. Wundt Block Illusion [Java]
  1759. Y. Sawayama's Lemma [Java]
  1760. Y. Sawayama's Theorem
  1761. YATPOPT, Yet Another Trigonometric Proof of the Pythagorean Theorem
  1762. Yet Another Appearance of the Golden Ratio [JavaScript, GeoGebra]
  1763. Yet Another Concurrence on the 9-Point Circle [JavaScript, GeoGebra]
  1764. Yet Another Seven Circles Theorem [JavaScript, GeoGebra]
  1765. Zaslavsky's theorem [Java]
  1766. Zöllner's Illusion [Java]

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