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

    An Inequality with Angle Trisectors

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

    Rotating Square in Search of the Golden Ratio

  1300. Rotating Top Illusion [Java]
  1301. Rotating Gears [Java]
  1302. Rotation Transform [Java]
  1303. Rouse Ball's Fallacy [Java]
  1304. Rusty Compass Construction of Equilateral Triangle [Java]
  1305. Salinon: From Archimedes' Book of Lemmas [Java]
  1306. Sam Loyd's Geometric Puzzle
  1307. Sam Loyd's Son's Dissection [Java]
  1308. Sanchez's Areas in Bottema's Configuration [JavaScript]
  1309. Sangaku: Reflections and the Phenomenon
  1310. Sangaku: Critique of My View and a Response
  1311. Sangaku and The Egyptian Triangle
  1312. Sangaku in a Square
  1313. Sangaku Iterations, Is it Wasan? [Java]
  1314. Sangaku via Peru
  1315. Sangaku with 8 Circles
  1316. Sangaku with Angle between a Tangent and a Chord [Java]
  1317. Sangaku with Quadratic Optimization
  1318. Sangaku with Three Mixtilinear Circles [Java]
  1319. Sangaku with Versines [Java]
  1320. Sangakus with a Mixtilinear Circle [Java]
  1321. Scintillating Grid Illusion [Java]
  1322. Schwarz Lantern [Java]
  1323. Secant Angles [Java]
  1324. Secant Angles II [Java]
  1325. Secant, Normal, Tangent [Java]
  1326. Secant, Tangents and Orthogonality [Java]
  1327. See-Saw Lemma [Java]
  1328. Segment Projections in Equilateral Triangle
  1329. Segment Trisection [Java]
  1330. Segment Trisection II
  1331. Segment Trisection Induced by Parallels to Medians [Java]
  1332. Semicircle on Square [JavaScript]
  1333. Semi-regular Tessellation on Hinges A [Java]
  1334. Semi-regular Tessellation on Hinges B [Java]
  1335. Separating point sets with a circle
  1336. Sequences of Concyclic Points on a Conic
  1337. Sequences of Touching Circles
  1338. Serendipitous Beauty [Java]
  1339. Serendipitous Proofs Of The Pythagorean Theorem
  1340. Seven and the Eighth Circle Theorem
  1341. Seven Circles Theorem [Java]
  1342. Seven Concyclic Points [Java, GeoGebra]
  1343. Seven Equal Circles [JavaScript, GeoGebra]
  1344. 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
  1345. Shaggy Dog Theorem
  1346. Shapes in a lattice
  1347. Shapes of constant width [Java]
  1348. Shearing and Translation in Pythagorean Pants [Java]
  1349. Shearing a Polygon into a Triangle of Equal Area [Java]
  1350. Shearing Butterflies in Quadrilaterals [Java]
  1351. Shedding Light on the Ball for Eyeballing
  1352. Shepard's Parallelogram Illusion [Java]
  1353. Shifted by the Inradius Tangent Collinearities [JavaScript, GeoGebra]
  1354. Shifting Cube Illusion [Java]
  1355. Shimmer Illusion [Java]
  1356. Short Construction of the Geometric Mean
  1357. Shortest Fence in a Quarter-Circle Pasture
  1358. Shredding the torus
  1359. Sides and Area of Pedal Triangle
  1360. Sierpinski Gasket By Common Trema Removal [Java]
  1361. Sierpinski Gasket Via Chaos Game [Java]
  1362. Sierpinski Gasket by Trema Removal [Java]
  1363. Similar Parallelograms on Sides of a Triangle
  1364. Similar Triangles And More
  1365. Similar Triangles in Crossing Circles [JavaScript, GeoGebra]
  1366. Similar Triangles on Sides of a Quadrilateral [Java]
  1367. Similar Triangles on Sides and Diagonals of a Quadrilateral [Java]
  1368. Similarity Is About Proportion - And Ways to Look at It
  1369. Simple dissection of an acute triangle
  1370. Simple Property of Circle Through the Incenter [JavaScript]
  1371. Simple Quadrilaterals Tessellate the Plane [Java]
  1372. Simson Line [Java]
  1373. Simsons and 9-Point Circles in Cyclic Quadrilateral [Java]
  1374. Simultaneous Contrast Illusion [Java]
  1375. Simultaneous Contrast Illusion II [Java]
  1376. Simultaneous Contrast: Koffka's Illusion [Java]
  1377. Simultaneous Contrast: White's Illusion [Java]
  1378. Simultaneous Contrast: White's Illusion, V-Variant [Java]
  1379. Three Concurrent Chords at 60 Degrees Angles
  1380. Simultaneous Diameters in Concurrent Circles [Java]
  1381. Simultaneous Generalization of the Theorems of Ceva and Menelaus
  1382. Sine and Cosine of 15 Degrees Angle
  1383. Sine, Cosine, and Ptolemy's Theorem
  1384. Six Circles with Concurrent Pairwise Radical Axes
  1385. Six Circles Theorem (Bui Quang Tuan) [Java]
  1386. Six Circles Theorem (Elkies) [Java]
  1387. Six Circles Theorem (Evelyn) [Java]
  1388. Six Concyclic Circumcenters in Fermat's Configuration [JavaScript, GeoGebra]
  1389. Six Concyclic Points [Java]
  1390. Six Concyclic Points on Sides of a Triangle
  1391. Six Concyclic Points II [Java]
  1392. Six Concyclic Points via Antipedal Triangle [JavaScript, GeoGebra]
  1393. Six Incircles in an Equilateral Triangle [Java]
  1394. Six Point Circle [Java]
  1395. SixPoints, ThreeLines [Java]
  1396. Soddy Circles and David Eppstein's Centers [Java]
  1397. Some Properties of Common Tangents [JavaScript, GeoGebra]
  1398. Some Properties of Napoleon's Configuration [Java]
  1399. Spectral Decomposition of a Planar Quadrilateral
  1400. Sperner's Lemma [Java]
  1401. Spiral Similarity [Java]
  1402. Spiral Similarity and CollinearitySpiral Similarity and Collinearity [Java, GeoGebra]
  1403. Spiral Similarity Leads to Equilateral Triangle [JavaScript, GeoGebra]
  1404. Square and Circle in a Gothic Cupola
  1405. Square from Four Points, One on Each Side
  1406. Square From Nowhere [Java]
  1407. Square in a Circle Inscribed in a Square [Java]
  1408. Square in a Right Triangle [Java]
  1409. Square in Chair [Java]
  1410. Square Inscribed in a Triangle [Java]
  1411. Square Inscribed in a Triangle II [Java]
  1412. Square, Similarity and Slopes [Java]
  1413. Squares and Straight Tetrominoes
  1414. Squares in Circles [Java]
  1415. Squares in Semicircle and Circle
  1416. Squares in Triangles
  1417. Squares Inscribed In a Triangle III [Java]
  1418. Squares on Sides of a Parallelogram [JavaScript, GeoGebra]
  1419. Squares on Sides of a Quadrilateral [Java]
  1420. Squares on Sides of a Quadrilateral II [JavaScript, GeoGebra]
  1421. Square out of a Quadrilateral
  1422. Squaring a circle
  1423. Squaring a Rectangle [Java]
  1424. Squinting Eyes Theorem [Java]
  1425. SSA
  1426. SSS
  1427. Staircase Illusion [Java]
  1428. Star of David [Java]
  1429. Stathis Koutras' Butterfly
  1430. Stathis Koutras' Theorem
  1431. Stathis Koutras' Theorem II
  1432. Steiner's Circumellipse in Barycentric Coordinates [GeoGebra]
  1433. Steiner's Inellipse
  1434. Steiner's Sangaku
  1435. Steiner's Ratio Theorem
  1436. Steiner's Theorem
  1437. Stereographic Projection [Java]
  1438. Stereographic Projection and Inversion
  1439. Stereographic Projection and Radical Axes
  1440. Stewart's theorem
  1441. Straight Edge Only Construction of Polar
  1442. Structural Constellation [Java]
  1443. Sums of Squares in Circle
  1444. Sum of Squares in Equilateral Triangle
  1445. Sum of Squares of Distances to Vertices
  1446. Sum of Two Squares [JavaScript, GeoGebra]
  1447. Sums of Unit Vectors
  1448. Surprising 60° in Equilateral Triangle [JavaScript, GeoGebra]
  1449. Sylvester's Problem
  1450. Symmedians
  1451. Symmetries in a Triangle
  1452. Symmetries in Triangle II [JavaScript, GeoGebra]
  1453. Symmetry in an Asymmetric Configuration [JavaScript, GeoGebra]
  1454. (tan(72°)±tan( 60°)sin(30°$\mp$18°)= 1
  1455. Tangency Point of Two Circles [Java]
  1456. Tangent and Secant [Java]
  1457. Tangent as a Radical Axis [Java]
  1458. Tangent Circles, Similarity and Equal Segments [JavaScript, GeoGebra]
  1459. Tangent Circles and an Isosceles Triangle [Java]
  1460. Tangent Circles in a Parallelogram [Java]
  1461. Tangent, Cotangent and Square Roots Inequality
  1462. Tangent Curiosity in Equilateral Triangle [JavaScript, GeoGebra]
  1463. Tangent in Concurrency [Java]
  1464. Tangent Lines and Circles in Convex Quadrilateral [Java]
  1465. Tangent of 22.5° - Proof Wthout Words
  1466. Tangent to Circle in Three Steps
  1467. Tangent to the Incircle [JavaScript]
  1468. Tangent to two circles
  1469. Tangents And Bisectors
  1470. Tangents and Diagonals in Cyclic Quadrilateral [Java]
  1471. Tangents, Perpendiculars and Geometric Mean [Java]
  1472. Tarski-Banach Decompositions
  1473. Taylor circle [Java]
  1474. Terquem's Theorem [Java]
  1475. Tesseract [Java]
  1476. Tetrahedron in Parallelepiped
  1477. Tetrahedron with Equiareal Faces
  1478. Tetrahedron with Equiareal Faces, II
  1479. Tetrahedron with Orthogonal Perpendiculars to Opposite Edges
  1480. Thales' Theorem [Java, GeoGebra]
  1481. Thales' Theorems
  1482. Thales on Angle Bisectors [JavaScript, GeoGebra]
  1483. The 80-80-20 Triangle
  1484. The Beauty of Fractions
  1485. The Center of Gravity and Motion of Material Points [JavaScript, GeoGebra]
  1486. The Chaos Game: Address Space vs IFS [Java]
  1487. The Devil is in the Exceptions
  1488. The Eutrigon Theorem [Java, GeoGebra]
  1489. The Golden Pentacross
  1490. The Law of Cosines by Larry Hoehn
  1491. The Midpoint of an Angle Bisector - a Problem by A. A. Zaslavsky
  1492. The only rational triangle
  1493. The Pythagorean Theorem from a Combinatorial Identity
  1494. The Schwarz Lantern Explained [JavaScript]
  1495. The Shape of the Colosseum [Java, GeoGebra]
  1496. The Shortest Crease [JavaScript, GeeoGebra]
  1497. Thébault's Problem I [Java]
  1498. Thébault's Problem II [Java]
  1499. Thébault's Problem III [Java]
  1500. Theorem of Complete Quadrilateral [Java]
  1501. Theorem of Three Tangents to a Conic [JavaScript, GeoGebra]
  1502. Theorems of Ceva and Menelaus, an Illustrated Generalization [Java]
  1503. There is no Difference Between Equilateral Triangles
  1504. Third Unusual Identity in Triangle
  1505. Three Circles and Area [JavaScript]
  1506. Three Circles and Common Tangents
  1507. Three Circles through the Incenter
  1508. Three Collinear and Four Concyclic Points [Java, GeoGebra]
  1509. Three Concurrent Circles [Java]
  1510. Three Common Chords in Three Concurrent Circles [Java, GeoGebra]
  1511. Three Congruent Circles by Reflection [Java]
  1512. Three Congruent Circles by Reflection II [Java]
  1513. Three Congruent Circles by Reflection III [Java]
  1514. Three Equal Circles in a Semicircle
  1515. Three Euler Lines That Are Four [JavaScript]
  1516. Three Incircles In a Right Triangle [Java]
  1517. Three Incircles in a Triangle [Java]
  1518. Three Isosceles Triangles [Java]
  1519. Three Orthogonal Circles Through Three Given Points [Java]
  1520. Three lines at 60 degres and Circumcenters [JavaScript, GeoGebra]
  1521. Three pairs of Parallel Lines and a Conic [JavaScript, GeoGebra]
  1522. Three Parabolas with Common Directrix [JavaScript,GeoGebra]
  1523. Three Parallels in a Triangle [Java]
  1524. Three Points on a Parabola [Java]
  1525. Three Pyramids are Better Than Two [Java]
  1526. Three regions sharing a common boundary
  1527. Three Similar Polygons [Java]
  1528. Three Similar Triangles [Java]
  1529. Three Similar Triangles II [Java]
  1530. Three Squares and Two Ellipses [Java]
  1531. Three Tangent Circles [Java]
  1532. Three Tangents, Three Chords in Ellipse [Java]
  1533. Three Tangent Circles Sangaku
  1534. Three Tangents Theorem [Java]
  1535. Three Tangents, Three Secants [Java]
  1536. Three Touching Circles [Java]
  1537. Three Travelers in Uniform Motion
  1538. Three visions of a fact [GeoGebra]
  1539. Tiling a Chessboard with Dominoes
  1540. Tiling Sphere by Four Equal Triangles
  1541. Toothpick Construction of a Square [Java]
  1542. Topological preliminaries
  1543. Torque as a Mathematical Object
  1544. Touching Circles [Java]
  1545. Touching Circles and Concurrency [Java]
  1546. Touching Circles with Given Centers II [Java]
  1547. Train wheel quandary
  1548. Transitivity in Action [Java]
  1549. Translated Triangles [Java]
  1550. Translation Transform [Java]
  1551. Transversal Trigonometry [JavaScript, GeoGebra]
  1552. Triangle ABC is right iff sin²A + sin²B + sin²C = 2
  1553. Triangle Area in Square with Incircle
  1554. Triangle Areas in a Parallelogram [Java]
  1555. Triangle Areas in a Parallelogram II [Java]
  1556. Triangle by HM segments [JavaScript, GeoGebra]
  1557. Triangle Classification [Java]
  1558. Triangle constructions and facts [Java]
  1559. Triangle Construction from Angle, Altitude and Median
  1560. Triangle from a, A, mc/mb
  1561. Triangle from Angle Bisector, Altitude and Circumradius
  1562. Triangle from Angle Bisector, Altitude and Inradius
  1563. Triangle from Inradius, Circumradius, Side or Angle
  1564. Triangle from Median, Altitude and Circumradius
  1565. Triangle from Side, Circumradius, and Median
  1566. Triangle from Angle, Inradius, and Difference of Sides
  1567. Triangle from Side, Inradius, and Altitude
  1568. Triangle from Side, Median, and Altitude
  1569. Triangle from Two Side Lines and the Euler Line
  1570. Triangle in a 4x4 Square
  1571. Triangle in a 5x5 Square
  1572. Triangle of Maximum Area
  1573. Triangle of Maximum Area II
  1574. Triangle of Minimum Perimeter [JavaScript]
  1575. Triangle Sliding on Two Straight Lines [Java]
  1576. Triangles, Squares and Areas from Temple Geometry [Java]
  1577. Triangles on HO
  1578. Triangles with Equal Area [Java]
  1579. Triangles with Sides in Geometric Progression [JavaScript]
  1580. Triangular Areas in Circle [JavaScript, GeoGebra]
  1581. Triangular Billiard [Java]
  1582. Triangulating Squares
  1583. Triangle with a 60 degrees angle [JavaScript, GeoGebra]
  1584. Trigonometry by Watching
  1585. Trigonometric Form of Ceva's Theorem [Java]
  1586. Trigonometric Functions [Java]
  1587. Trigonometric Identities with Arctangents
  1588. Trigonometric Proof of the Pythagorean Theorem
  1589. Trigonometry According to Ptolemy
  1590. Trigonometry by Paper Folding
  1591. Trisect Segment: 2 Circles, 4 Lines
  1592. Trisectable angle that is not constructible
  1593. Tucker Circles [Java]
  1594. Tucker Circles Through Homothety [Java]
  1595. Twelve Matchsticks Area Puzzle
  1596. Twin Segments in Four Semicircles
  1597. Twins Beget Twins
  1598. Twisted Cord Illusion [Java]
  1599. Two Altitudes, One Midpoint
  1600. Two Arbelos, Two Chains
  1601. Two Cevians and Proportions in a Triangle
  1602. Two Circles, Two Segments - One Ratio
  1603. Two Circles and a Limit - Analytic Proof
  1604. Two Circles and One More [Java]
  1605. Two Circles and Two More [Java]
  1606. Two Circles, Ellipse, and Parallel Lines
  1607. Two Circles in a Parallelogram [Java]
  1608. Two Circles Inscribed in a Parallelogram [Java]
  1609. Two Circles in a Square [Java]
  1610. Two Circles in a Square II [Java]
  1611. Two Circles in an Angle [Java]
  1612. Two Circles on Angle Bisector [Java]
  1613. Two Circles on a Side of a Triangle [Java]
  1614. Two Circles, Tangents and Two Collinearities [JavaScript, GeoGebra]
  1615. Two Circumcircles And Two Pairs of Parallels [Java]
  1616. Two Circumcircles in Triangle [Java]
  1617. Two Conditions for a Triangle to Be Equilateral
  1618. Two coins puzzle
  1619. Two Congruent Circles by Reflection [Java]
  1620. Two Conics, Pascal, Chasles, and Cross-Ratio [JavaScript, GeoGebra]
  1621. Two Color Coloring of the Plane [Java]
  1622. Two Common Chords in Three Concurrent Circles [Java, GeoGebra]
  1623. Two Diameters and Longest Common Chord [Java, GeoGebra]
  1624. Two Equilateral Triangles [Java]
  1625. Two Homotheties in a Parallelogram [Java]
  1626. Two Intersecting Circles [Java]
  1627. Two Lines - Two Circles [Java]
  1628. Two Pencils of Parallel Lines [JavaScript, GeoGebra]
  1629. Two Pairs of Parallel Lines in a Triangle [Java, GeoGebra]
  1630. Two Parallels in a Triangle and One More [Java]
  1631. Two Pascals Merge into One [JavaScript, GeoGebra]
  1632. Two Perpendiculars From a Point to a Line [Java]
  1633. Two Planar Constructions Related to a 3D Problem
  1634. Two Perspective Triangles
  1635. Two Quadrilaterals [Java]
  1636. Two Quadruplets of Concyclic Points [JavaSCript,GeoGebra]
  1637. Two Quadruplets of Tangent Circles [Java]
  1638. Two Refinements of Ionescu-Weitzenbock Inequality
  1639. Two Regular Pentagons Joined at Vertex
  1640. Two Related Triangles of Equal Areas
  1641. Two Sangaku with Equal Incircles
  1642. Two Similar Triangles with Parallel Sides [JavaScript, GeoGebra]
  1643. Two Simsons in a Triangle [Java]
  1644. Two Squares and Another Square [Java]
  1645. Two Squares In Circle [JavaScript]
  1646. Two Symmetric Triangles Are Directly Decomposable [Java]
  1647. Two Touching Circles [Java]
  1648. Two Triangles Inscribed in a Conic [Java, GeoGebra]
  1649. Two Triangles Inscribed in a Conic - with Elementary Solution [Java, GeoGebra]
  1650. Two Triangles Inscribed in a Conic - with Solution [Java, GeoGebra]
  1651. Two Triangles With Common Base and Altitude [Java]
  1652. Two Triangles Sharing the Circumcenter
  1653. Two Triples of Concurrent Cevians and a Conic [JavaScript, GeoGebra]
  1654. Two Triples of Concurrent Circles [Java]
  1655. Two Triples of Similar Triangles [Java]
  1656. Uncommon Triangle Constructions
  1657. Unfolding Spirals Illusion [Java]
  1658. Urquhart's Theorem
  1659. USA 1975 Problem 4
  1660. Use of Homogeneous Coordinates for Geometrical Calculations
  1661. Van Khea's Fourth Identity in Triangle
  1662. Van Obel Theorem, Determinants, and Barycentric Coordinates
  1663. Van Schooten's and Pompeiu's Theorems [Java]
  1664. Van Schooten's Locus Problem [Java]
  1665. Variations on the Theme of Tremas [Java]
  1666. Variations on the Theme of Tremas II [Java]
  1667. Varignon Parallelogram [Java]
  1668. Various Geometries
  1669. Vecten-Nikolin Hexagon [Java]
  1670. Vecten's Collinearity [Java]
  1671. Vecten's Mesh [Java]
  1672. Vector Addition and Subtraction [Flash]
  1673. Vector Algebra in Tetrahedron
  1674. Vector product
  1675. Versatile Theorem [Java]
  1676. Vertical and Horizontal Dominoes on a Chessboard
  1677. Vietnamese Extension of a Japanese Theorem [JavaScript]
  1678. Vietnamese Lemma towards an Extension of a Japanese Theorem [JavaScript, GeoGebra]
  1679. Viewing a Statue
  1680. Viviani's Theorem [Java]
  1681. Viviani's Theorem II [Java]
  1682. Viviani in Isosceles Triangle [Java]
  1683. Viviani by Inversion
  1684. Viviani by Vectors
  1685. Viviani's 3D Analogue
  1686. Sanchez's Viviani's Area Analogue [JavaScript, GeoGebra]
  1687. Volume and Area of Torricelli's Trumpet [JavaScript]
  1688. Volume of Fibonacci Tetrahedron
  1689. Volume of Sphere and Volume of an Ellipsoid
  1690. Volumes of Two Pyramids
  1691. Wallace-Bolyai-Gerwien Theorem
  1692. Wallace's Theorem [Java]
  1693. Watt's and Chebyshev's Linkage [Java]
  1694. Weird curves bound normal areas
  1695. Weitzenbock by Sanchez
  1696. Weitzenböck Inequality [Java]
  1697. What does one see inside a spherical mirror? [Java]
  1698. What Is Analytic Geometry?
  1699. What Is Angle?
  1700. What Is Angle Chasing?
  1701. What Is Area?
  1702. What Is Ellipse?
  1703. What Is Indirect Proof?
  1704. What Is Line?
  1705. What Is Point?
  1706. When a Triangle is A-cute
  1707. When a Triangle is Equilateral [Java]
  1708. When Is Triangle Equilateral: Marian Dinca's Criterion
  1709. When Is Triangle Isosceles: Miguel Ochoa Sanchez's Criterion
  1710. When Two Segments Intersect? [JavaScript]
  1711. When index equals content [Java]
  1712. Wonderful Trigonometry In Equilateral Triangle
  1713. Why a Diameter Is the Longest Chord?
  1714. William Wallace's Proof of the Butterfly Theorem
  1715. William Wallace's 1803 Statement of the Butterfly Theorem [JavaScript, GeoGebra]
  1716. Wittenbauer's Parallelogram [Java]
  1717. Wundt Illusion [Java]
  1718. Wundt Block Illusion [Java]
  1719. Y. Sawayama's Lemma [Java]
  1720. Y. Sawayama's Theorem
  1721. YATPOPT, Yet Another Trigonometric Proof of the Pythagorean Theorem
  1722. Yet Another Appearance of the Golden Ratio [JavaScript, GeoGebra]
  1723. Yet Another Concurrence on the 9-Point Circle [JavaScript, GeoGebra]
  1724. Yet Another Seven Circles Theorem [JavaScript, GeoGebra]
  1725. Zaslavsky's theorem [Java]
  1726. Zöllner's Illusion [Java]

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