Geometry articles, theorems, problems

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Note that in the list below the pages that carry Java applets are designated with (*).

  1. 1 + 27 = 12 + 16 Sangaku
  2. 120° Breeds 90° (*)
  3. 3 Roads, 3 Travelers (*)
  4. 3 Utilities Puzzle
  5. 3D Quadrilateral - a Coffin Problem
  6. 3-4-5 Triangle by a Kid
  7. 4 Travelers problem
  8. 6 to 9 Point Circle (*)
  9. 60o Breeds 90o (*)
  10. 7 = 2 + 5 Sangaku
  11. 9-point Circle as a Locus of Concurrency (*)
  12. A better solution to a difficult sangaku problem
  13. A Case of Similarity (*)
  14. A Circle in an Isosceles triangle from the 2006 Itish MO
  15. A Circle Related to Incenter and Circumcenter (*)
  16. A Circle Rolling in an Equilateral Triangle (*)
  17. A Circle With Two Centers (*)
  18. A Geometric Limit
  19. A Hard but Important Sangaku
  20. A Lemma of Equal Areas (*)
  21. A Line in a Square Grid (*)
  22. A Line in Triangle Through the Circumcenter (*)
  23. A Parallelogram in Triangle (*)
  24. A Possibly First Proof of the Concurrence of Altitudes
  25. A Problem In an Isosceles Right Triangle (*)
  26. A Problem on an Icy Cone
  27. A problem with equilateral triangles (*)
  28. Properties of the Figures in Euclid I.47
  29. A Property of Equiangular Polygons (*)
  30. A Property of Rhombi (*)
  31. A Property of the Line IO (*)
  32. A Property of the Line IO: A Bundled Statement and Proof (*)
  33. Property of the Line IO: a proof from the Book (*)
  34. A Relation in Triangle
  35. A Restored Sangaku
  36. A Sangaku Follow-Up on an Archimedes' Lemma (*)
  37. A Sangaku: Two Unrelated Circles (*)
  38. A Sangaku with an Egyptian Attachment (*)
  39. A Sangaku with Many Circles and Some (*)
  40. A Stronger Triangle Inequality (*)
  41. A Sushi Morsel
  42. A Triangle of Antreas Hatzipolakis (*)
  43. A Trigonometric Identity in a Right Triangle from the 1960 IMO
  44. A Trigonometric Solution to a Difficult Sangaku Problem
  45. About a Line and a Triangle (*)
  46. Abul Wafa al-Buzjani's Proof of the Pythagorean Theorem (*)
  47. Adams' Circle (*)
  48. Addition and Subtraction Formulas for Sine and Cosine
  49. Addition and Subtraction Formulas for Sine and Cosine II
  50. All Triangles Are Isosceles (*)
  51. Altitudes (*)
  52. Altitudes and the Euler Line (*)
  53. Altitudes and the Power of a Point (*)
  54. An Extension of van Schooten's Theorem
  55. An Identity in a Regular Heptagon
  56. An Isoperimetric Theorem (*)
  57. An Old Japanese Theorem
  58. Angle: An Illustrated Classification (*)
  59. Angle Bisector in Equilateral Trapezoid
  60. Angle Bisector in Parallelogram (*)
  61. Angle Bisector in Touching Circles (*)
  62. Angle Bisectors (*)
  63. Angle Bisectors and Perpendiculars in a Quadrilateral (*)
  64. Angle Bisectors in Ellipse (*)
  65. Angle Bisectors in Ellipse II (*)
  66. Angle Bisectors in a Quadrilateral (*)
  67. Angle Bisectors On Circumcircle (*)
  68. Angle Subtended by a Diameter (*)
  69. Angle Preservation Property (*)
  70. Angle Trisection by Archimedes (*)
  71. Angle Trisection by Hippocrates (*)
  72. Angle Trisection by Paul Vjecsner (*)
  73. Angle Trisectors on Circumcircle (*)
  74. Angle Trisection by Paper Folding
  75. Angles in a Cube I (*)
  76. Angles in a Cube II (*)
  77. Angles in a Cube III (*)
  78. Angles in Triangle Add to 180o
  79. Anticomplementary Triangle Surprise (*)
  80. Antiparallel (*)
  81. APMO 1995 Problem 3
  82. APMO 1998, Problem 4
  83. APMO 2002, Problem 3
  84. APMO 2003, Problem 2
  85. APMO 2004, Problem 2
  86. Apollonian Circles Theorem (*)
  87. Apollonian Gasket (*)
  88. Apollonius Problem (*)
  89. Approximate Construction of Regular Pentagon by A. Dürer
  90. Arbelos -- the Shoemaker's Knife (*)
  91. Archimedes' Book of Lemmas
  92. Archimedes' Law of the Lever (*)
  93. Archimedes' Method
  94. Archimedes' Twins in the Edo Period
  95. Area of Median Triangle (*)
  96. Area of Parallelogram (*)
  97. Area of the Union of Two Squares (*)
  98. Area of Triangle (*)
  99. Areas and Centroid in a Triangle (*)
  100. Arithmetic-Geometric Mean Inequality
  101. Arithmetic Mean Sangaku (*)
  102. Arithmetic Mean Sangaku II (*)
  103. Around the Incircle (*)
  104. Around the Incircle in a Polygon (*)
  105. Artisans vs Mathematicians
  106. Asian Pacific Kite (*)
  107. Assimilation Illusion (*)
  108. Asymmetric Propeller (*)
  109. Barbier, The Theorem of (*)
  110. Barycentric coordinates
  111. Bender: A Visual Illusion (*)
  112. Between Major and Minor Circles (*)
  113. Bevan's Point and Theorem (*)
  114. Bimedians in a Regular Tetrahedron (*)
  115. Bisectal Circle (*)
  116. Bisecting arcs
  117. Bisecting a shape
  118. Bisector of an imaginary angle may be real
  119. Bottema Shatters Japan's Seclusion (*)
  120. Bottema's theorem (*)
  121. Bottles in a Wine Rack (*)
  122. Bounded Distance
  123. Brahmagupta's Theorem (*)
  124. Brianchon in Ellipse (*)
  125. Brianchon's theorem (*)
  126. Brick Wall Illusion (*)
  127. Bride's Chair (*)
  128. Brief introduction to Rational Trigonometry
  129. Broken Chord Theorem (*)
  130. Building a Bridge (*)
  131. Building Bridges (*)
  132. Bulging lines illusion (*)
  133. Butterfly Theorem
  134. Cabart's Collinearity
  135. Cantor Set and Function
  136. Cantor's Theorem (*)
  137. Carnot's Theorem
  138. Carnot's Theorem
  139. Carnot's Theorem (Generalization of Wallace's theorem) (*)
  140. Carpet With a Hole (*)
  141. Carpets Theorem (*)
  142. Centroid, a Characteristic Property Of
  143. Centroids in a Polygon (*)
  144. Chasing Angles in Pascal's Hexagon (*)
  145. Circle, Isosceles Triangle and Fixed Point (*)
  146. Circle in a Square Inscribed in a Circle (*)
  147. Circles on the sides of a triangle from the 2005 Austrian MO
  148. Circles Tangent to Circumcircle (*)
  149. Circles with Tangent Diameters (*)
  150. Ceva and Menelaus Meet on the Roads
  151. Ceva in Circumscribed Quadrilateral
  152. Ceva's Theorem
  153. Ceva's Theorem and Fibonacci Bamboozlement (*)
  154. Cevian Nest (*)
  155. Cevian Triangle (*)
  156. Chain of Inscribed Circles (*)
  157. Chaos, Emergence of (*)
  158. Chasles' Theorem
  159. Cherchez le quadrilatère cyclique (*)
  160. Cherchez le quadrilatère cyclique II (*)
  161. Chords, Concurrency and Orthic Triangle (*)
  162. Chvatal's Art Gallery Theorem (*)
  163. Circle and Hyperbola as Lighthouse Curves
  164. Circle and Hyperbola as Lighthouse Curves II
  165. Circle Crosses Triangle (*)
  166. Circle Concurrency and Spiral Similarity (*)
  167. Circle Division by Euclid and Leonardo (*)
  168. Circle of Similitude (*)
  169. Circle through the Incenter (*)
  170. Circle through the Incenter And Antiparallels (*)
  171. Circle-Stacking Theorem (*)
  172. Circles And Parallels (*)
  173. Circles and Semicircles in Rectangle (*)
  174. Circles in a Circular Segment
  175. Circles in Morley's Triangles (*)
  176. Circles Lined on the Legs of a Right Triangle (*)
  177. Circles On Cevians (*)
  178. Circles through the Orthocenter (*)
  179. Circles with Equal Collinear Chords (*)
  180. Circular Poggendorff Illusion (*)
  181. Circle Inscribed in a Circular Segment (*)
  182. Circumcenter on Angle Bisector (*)
  183. Circumcevian Triangle (*)
  184. Classification of Quadrilaterals (*)
  185. Cleavance Center (*)
  186. Cleaver (*)
  187. Clifford's Chain (*)
  188. Coaxal Circles on Perpendicular Bisector (*)
  189. Coaxal Circles Theorem (*)
  190. Collage Theorem (*)
  191. Collinearity in Bicentric Quadrilaterals (*)
  192. Color Cycling on the Mandelbrot Set (*)
  193. Common Chord and a Tangent (*)
  194. Common Tangents to Two Circles I (*)
  195. Common Tangents to Two Circles II (*)
  196. The Complete Quadrilateral (*)
  197. Complex Numbers and Geometry
  198. Concurrency in the Intouch Triangle (*)
  199. Concyclic Circumcenters: A Dynamic View (*)
  200. Concyclic Points in a Triangle
  201. Concyclic Points in Bride's Chair (*)
  202. Concyclic Points in Equilateral Bumps (*)
  203. Conic Sections
  204. Conic Sections as Loci of Points (*)
  205. Conjugate Diameters in Ellipse (*)
  206. Consecutive Isosceles Decomposition (*)
  207. Constant Chord
  208. Constructing a triangle from its angle bisectors is in general impossible
  209. Construction of a Square from the 2008 Austrian MO
  210. Construction of a Triangle from the Circumcenter, Orthocenter and Incenter
  211. Construction of a Triangle from Two Vertices and the Centroid
  212. Construction of Conics from Pascal's Theorem (*)
  213. Construction of Regular Pentagon by H. W. Richmond
  214. Construction of the Angle Bisector (*)
  215. Construction of the Perpendicular Bisector (*)
  216. Construction problem (*)
  217. Constructions Related To An Inaccessible Point
  218. Construction with the Compass Only
  219. Continuous Line Illusion (*)
  220. Convex Polygon Is the Intersection of Half Planes (*)
  221. Copernicus' Theorem (*)
  222. Construction with Optimization in Angle - Problem the 1979 USA Mathematical Olympiad
  223. cos(36°) = (1 + 5)/4
  224. Cosine Law
  225. Counting Triangles and Segments in a Polygon (*)
  226. Cross-Ratio
  227. Crossed-Lines Construction of Shapes of Constant Width (*)
  228. Crossing Number of a Graph
  229. Cube Counting Ambiguity (*)
  230. Curry's Paradox (*)
  231. Cut the Cone (*)
  232. Cut the Cube (*)
  233. Cut the Cylinder (*)
  234. Cutting Squares
  235. Cutting Triangles off Regular Hexagon
  236. Cyclic Hexagon (*)
  237. Cycloids (*)
  238. Day's Sine Illusion (*)
  239. Delboeuf Illusion (*)
  240. Delian Problem Solved (*)
  241. Desargues in the Bride's Chair (with Pythagoras)
  242. Desargues' Theorem (*)
  243. Determination of p
  244. Diagonal Count (*)
  245. Diagonals in a Cyclic Quadrilateral
  246. Diameters and Chords (*)
  247. Directly Similar Figures (*)
  248. Disappearing Lines puzzle (*)
  249. Discovery of Duane DeTemple (*)
  250. Dissection of a 10×13 Rectangle into Two Chessboards
  251. Dissection of Cyclic Quadrilateral (*)
  252. Distance
  253. Distance between the Orthocenter and Circumcenter
  254. Distance Formula
  255. Distorted Lines Illusion (*)
  256. Divergent Spirals Illusion (*)
  257. Divide a Circle into N Parts of Equal Area
  258. Divide a segment into n equal parts
  259. Dividing Evenly a Quadrilateral (*)
  260. Dividing Evenly a Quadrilateral II (*)
  261. Double Generation Theorem (*)
  262. Drawing circles (*)
  263. Droz-Farny Circles (*)
  264. Droz-Farny Line Theorem (*)
  265. Duplication of the Cube
  266. Dynamic Construction of Ellipse and Other Curves (*)
  267. E. W. Dijkstra's Proof of the Pythagorean Theorem (*)
  268. Easy Construction of Bicentric Quadrilateral (*)
  269. Easy Construction of Bicentric Quadrilateral II (*)
  270. Egyptian Triangle By Paper Folding
  271. Eight Equal Tangents (*)
  272. Eight Point Circle (*)
  273. Ellipse in Arbelos (*)
  274. Ellipse Between Two Circles (*)
  275. Equal Angles in Two Circles (*)
  276. Equal Angles in Two Circles II (*)
  277. Equal Areas in Equilateral Triangle (*)
  278. Equal Areas In Parallelogram à la Pythagoras (*)
  279. Equal Chords in Crossing Circles (*)
  280. Equal Circles, Medial Triangle and Orthocenter (*)
  281. Equal Incircles Theorem (*)
  282. Equichordal points: Just Do It (*)
  283. Equidecomposition of a Rectangle and a Square (*)
  284. Equidecomposition of Two Rectangles (*)
  285. Equidecomposition of a Triangle and a Rectangle (*)
  286. Equidecomposition of a Triangle and a Rectangle II (*)
  287. Equidecomposition of Two Parallelograms (*)
  288. Equilateral and 3-4-5 Triangles (*)
  289. Equilateral Triangle on Parallel Lines (*)
  290. Equilateral Triangle, Straight Line and Tangent Circles (*)
  291. Equilateral Triangles and Incircles in a Square
  292. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral (*)
  293. Equilateral Triangles On Sides of a Parallelogram (*)
  294. Equilateral Triangles On Sides of a Parallelogram II (*)
  295. Equilateral Triangles on Sides of a Quadrilateral (*)
  296. Equilic Quadrilateral I, a Flawed Proof (*)
  297. Equilic Quadrilateral I, A Variation (*)
  298. Equilic Quadrilateral II (*)
  299. The Erdös-Mordell Inequality
  300. Escher's Theorem (*)
  301. Euclid's Fifth Postulate
  302. Euclid's Elements Reference Page
  303. Euclidean Construction of Center of Ellipse (*)
  304. Euclidean Construction of Tangent to Ellipse
  305. Euler Line Cuts Off Equilateral Triangle (*)
  306. Euler Line and 9-Point Circle (*)
  307. Euler's Formula and Poncelet Porism
  308. Existence of the Circumcenter (*)
  309. Existence of the Euler Line: An Elementary Proof (*)
  310. Experimentation with Dynamic Geometry Software: An Example (*)
  311. Euclid I.43 (*)
  312. Euclid I.43 Extended (*)
  313. Exterior Angle Theorem - an appreciation
  314. External Angle Bisector (*)
  315. Extouch Triangle in Poncelet Porism (*)
  316. Eyeball Theorem (*)
  317. Fagnano's problem (*)
  318. Faulty Symmetry
  319. Fermat point and 9-point Centers (*)
  320. Feuerbach's Theorem: A Proof (*)
  321. Find a Common Chord of Given Length (*)
  322. Find the Angle! (*)
  323. Finding Circle Center By Ruler Alone (*)
  324. Five Concyclic Points (*)
  325. Five Incircles in a Square (*)
  326. Five Incircles Theorem (*)
  327. Fixed Point in a Kite - Problem from the 1993 Asian Pacific MO
  328. Fixed Point of Circles Orthogonal to the Given One (*)
  329. Fixed points from 2007 Irish Mathematical Olympiad
  330. Fleury's Algorithm and Euler's Paths and Cycles (*)
  331. Foot of Altitude and Minimum Distance (*)
  332. Focal Definition of Ellipse (*)
  333. Focus and Directrix of Ellipse (*)
  334. For every object there is a distance at which it looks its best.
  335. Ford's touching circles
  336. Four Centroids and Parallels (*)
  337. Four Circles In a Triangle (*)
  338. Four Concurrent Lines in a Cyclic Quadrilateral (*)
  339. Four Construction Problems (*)
  340. Four Crossing Circles (*)
  341. Four Geometric Problems of Antiquity
  342. Four Hinged Squares (*)
  343. Four Incident Circles (*)
  344. Four Incircles in Equilateral Triangle (*)
  345. Four 9-Point Circles in a Quadrilateral (*)
  346. Four Pedal Circles (*)
  347. Four Touching Circles (*)
  348. Four Touching Circles II (*)
  349. Four Touching Circles III (*)
  350. Four Triangles, One Circle (*)
  351. Four Turtles (or Bugs, or Dogs, etc.) (*)
  352. Fractal curves and dimension (*)
  353. Friendly Kiepert's Perspectors (*)
  354. From Foci to a Tangent in Ellipse (*)
  355. Fundamental Theorem of 3-Bar Motion (*)
  356. Fuss' Theorem (*)
  357. Generalizing a Romanian Olympiad Problem (*)
  358. Geometric Illustration of a Convergent Series
  359. Geometric Mean Sangaku
  360. Geometric Meaning of the Geometric Mean
  361. Geometric Optimization from the Asian Pacific Mathematical Olympiad (*)
  362. Gergonne and Medial Triangles Are Orthologic (*)
  363. Gergonne and Soddy Lines Are Perpendicular (*)
  364. Gergonne in Ellipse (*)
  365. Gergonne and Van Obel Theorems
  366. Gion Shrine Problem
  367. Given Parabola, Find Axis
  368. Glide Reflection (*)
  369. Golden Ratio in Geometry
  370. Golden Ratio in an Irregular Pentagon
  371. Golomb's inductive proof of a tromino theorem (*)
  372. Gothic Arc (*)
  373. Graph Theory (*)
  374. Grebe: from Ladies' Diary to Carroll's Pillow (*)
  375. Grebe's Theorem
  376. Greg Markowsky's Problem for Parabola (*)
  377. Griffiths Points in Cyclic Quadrilateral (*)
  378. Half Turn, Reflection in Point (*)
  379. Halving a square
  380. Harmonic Mean Sangaku (*)
  381. Harmonic Ratio
  382. Harmonic Ratio in Complex Domain
  383. Hart Circle (*)
  384. Hart's Inversor (*)
  385. Haruki's Lemma (*)
  386. Haruki's Theorem
  387. Hausdorff Distance
  388. Hearty Munching on Cardioids (*)
  389. Height/Width Interplay Illusion (*)
  390. Hering Illusion (*)
  391. Hermann Grid Illusion (*)
  392. Heron's formula
  393. Heron's Formula: a Proof (*)
  394. Heron's Problem (*)
  395. Hexagon Parallel to Medial Triangle (*)
  396. Hexagon Parallel to Orthic Triangle (*)
  397. Hinged Squares (*)
  398. Hippocrates' Squaring of a Lune (*)
  399. Hippocrates' Squaring of Lunes
  400. Hjelmslev Theorem (*)
  401. Homologous Lines under Three Spiral Similarities (*)
  402. Homothety (*)
  403. Hooper's Paradox (*)
  404. Hunting Right Angles (*)
  405. Ian McGee's Observation (*)
  406. IMO 1998 Problem 1
  407. IMO 2004, Problem 5
  408. IMO 2006, Problem 1
  409. Impossible Fork (*)
  410. Impossible Frame (*)
  411. Importance of Occluders: Motion Binding (*)
  412. In the Wasan Spirit
  413. Incenters in Cyclic Quadrilateral (*)
  414. Incidence in Feuerbach's Theorem (*)
  415. Infinitude of Primes - A Topological Proof
  416. Inscribed and Central Angles in a Circle (*)
  417. Inscribed Angles (*)
  418. Inscriptible and Exscriptible Quadrilaterals (*)
  419. Inscriptible Quadrilateral: An Illustration (*)
  420. Internal Tangents to Three Circles (*)
  421. Intersecting Chords Theorem
  422. Intouch Triangle in Poncelet Porism (*)
  423. Invariant Area Sweep Implies Pythagorean Theorem (*)
  424. Inversion: Reflection in a Circle (*)
  425. Is Every Trapezoid Parallelogram? (*)
  426. Isosceles Tetrahedron
  427. Isogonal and Isotomic Conjugalities (*)
  428. Isogonal Concurrency (*)
  429. Isogonal image of the circumcircle (*)
  430. Isoperimetric Theorem and Inequality
  431. Isosceles on the Sides of a Triangle (*)
  432. Isotomy and Isogonality Hand-in-Hand (*)
  433. Iterations and the Mandelbrot Set (*)
  434. Iterations on a Circle Through Three Points (*)
  435. Iterations on Euler Lines (*)
  436. J. Casey's Additions to Euclid I.47
  437. Jan Stevens' proof of E. W. Dijkstra's Generalization of the Pythagorean theorem
  438. Joachimsthal's Notations
  439. John Sharp's Paradox (*)
  440. Join Circles by Given Segment (*)
  441. Joseph Keech in Bride's Chair (*)
  442. Judd Illusion (*)
  443. Julia sets indexing (*)
  444. Kanizsa Triangle (*)
  445. Kiepert's Theorem (*)
  446. Knots [avi movie]
  447. Kürschak's Tile and Theorem (*)
  448. La Hire's Theorem (*)
  449. Langman's Paradox (*)
  450. Lean Napoleon's Triangles (*)
  451. Length of the diagonal of the unit square equals the square root of 2
  452. Lepidoptera of the Circles (*)
  453. Line IO in Bicentric Quadrilaterals
  454. Line through a Point in an Angle
  455. Line, Circle, and Fixed Points (*)
  456. Locus of Points in a Given Ratio to Two Points (*)
  457. Longest segment
  458. MacLaurin's Construction of Conics (*)
  459. Maimed Cake (*)
  460. Malfatti's Problem (*)
  461. Malfatti's Problem, Hart's Solution (*)
  462. Mathematical Droodles (*)
  463. Mathematics in Pizzeria (*)
  464. MathPro Logo (*)
  465. Maximal Properties of the Pythagorean Relation
  466. Maximum Area Property of Equilateral Triangles
  467. Maximum Area Property of the Medial Triangle
  468. Maximum Perimeter Property of the Incircle {*}
  469. Maximum Ratio of Similarity - Belarus Olympiad Problem
  470. Maxwell's Theorem (*)
  471. May the Product of Planes Be a Sphere?
  472. Mechanical Proofs of the Pythagorean Theorem
  473. Median in Touching Circles (*)
  474. Medians
  475. Medians in a Quadrilateral (*)
  476. Menelaus Theorem (*)
  477. Metric Relations in a Triangle
  478. Midline in Similar Triangles (*)
  479. Midpoints from Gergonne Triangle (*)
  480. Midpoints of the Lines Joining In- and Excenters (*)
  481. Minesweeper Theorem
  482. Miquel's Point (*)
  483. Miquel's Point of a 4-line Via Spiral Similarity (*)
  484. Miquel's Theorem for Circles (*)
  485. Mirror on the Wall (*)
  486. Mirror property of the altitudes (*)
  487. Mirror Property of Altitudes via Pascal's Hexagram (*)
  488. Mixtilinear Circles and Concurrence (*)
  489. Moebius Strip [avi movie]
  490. Möbius' Strip (*)
  491. Models and Metamathematics
  492. Monge via Desargues (*)
  493. Monge via Desargues II (*)
  494. Morley Constellation (*)
  495. Morley's Miracle (*)
  496. Morphing (*)
  497. Müller-Lyer Illusion (*)
  498. Munching on Circles
  499. Munching on Inscribed Angles
  500. My logo
  501. Nagel Line (*)
  502. Nagel point (*)
  503. Nagel Point of the Medial Triangle (*)
  504. Nagel's Theorem (*)
  505. Napoleon's Theorem
  506. Necker Cube (*)
  507. Neuberg Sangaku (*)
  508. Newton's Construction of Conics (*)
  509. Newton's Theorem (*)
  510. Nine Point Circle (*)
  511. Nine Point Circle: an Elementary Proof (*)
  512. No Equilateral Triangles, Please (*)
  513. Nobbs' Points, Gergonne Line (*)
  514. Non-Eclidean Geometries
  515. No-Pedal Collinearity (*)
  516. Occluded Wheel Illusion (*)
  517. Olof Hanner's Jigsaw Puzzle (*)
  518. On the Difference of Areas
  519. Optimization Problem in Acute Angle (*)
  520. Octagon in Parallelogram: Overlapping Regions (*)
  521. Octagon in Parallelogram (*)
  522. Optical Property of Ellipse (*)
  523. Orbison illusion (*)
  524. Orthocenter and Three Equal Circles (*)
  525. Orthocenters
  526. Orthocenters of Two Triangles Sharing Circumcenter and Base (*)
  527. Orthocentric System From Rectangle (*)
  528. Orthodiagonal and Cyclic Quadrilaterals (*)
  529. Orthogonality in Isosceles Triangles (*)
  530. Orthologic Triangles in a Quadrilateral (*)
  531. Orthopole (*)
  532. Out of Pentagon Sangaku
  533. Packing shapes in a plane
  534. Pairs of Homologous Lines under Spiral Similarities (*)
  535. Pairs of Incircles in a Quadrilateral (*)
  536. Paper Folding & Cutting Sangaku
  537. Paper Folding Geometry
  538. Paper strip activities [avi movies]
  539. Paper strip activities, two strips
  540. Pappus' Generalization of Euclid I.47 (*)
  541. Pappus' Theorem (*)
  542. Parabola (*)
  543. Parabolic Mirror, Illustration (*)
  544. Parabolic Mirror, Theory (*)
  545. Paragon (*)
  546. Parahexagon (*)
  547. Parallel Chords (*)
  548. Parallel Chords II (*)
  549. Parallel Chords in Ellipse (*)
  550. Parallel Chords in Crossing Circles (*)
  551. Parallel Lines in a Quadrilateral (*)
  552. Parallelogram and Similar Triangles (*)
  553. Parallelogram Illusion (*)
  554. Parallelogram Iterations (*)
  555. Parallelogram Law (*)
  556. Parallelogram Law: A PWW (*)
  557. Parallelogram Law: A Tessellation (*)
  558. Partition of point sets in a plane
  559. Partitioning 3-Space with Circles
  560. Partitioning a Circle
  561. Pascal: Necessary and Sufficient (*)
  562. Pascal in a Cyclic Quadrilateral (*)
  563. Pascal Lines: Steiner and Kirkman Theorems (*)
  564. Pascal Lines: Steiner and Kirkman Theorems II (*)
  565. Pascal's theorem (*)
  566. Peacock's Tail Sangaku
  567. Pedal Collinearities (*)
  568. Pedal Polygons (*)
  569. Pedal Triangle and Isogonal Conjugacy (*)
  570. Pedoe's Theorem (*)
  571. Pencils of Cubics
  572. Pentagon, construction of
  573. Pentagon Proportions Sangaku
  574. Periodic patterns in sin(N) (*)
  575. Perpendicular Bisectors in an Inscriptible Quadrilateral (*)
  576. Perpendicular Bisectors in an Inscriptible Quadrilateral II (*)
  577. Perpendicular lines in a cyclic quadrilateral from 1975 Canadian Mathematical Olympiad
  578. Pivot Theorem (*)
  579. Plane Filling Curves (*)
  580. Plane Isometries
  581. Plato's Geometric Numbers: can you better Sam Loyd?
  582. Peaucellier Linkage (*)
  583. Pick's Theorem (*)
  584. Poggendorff's Illusion
  585. Point common to two similar rectangles (*)
  586. Point in a Square
  587. Points Generated by the Nine Points
  588. Polar Circle (*)
  589. Poles and Polars (*)
  590. Polygons: formality and intuition (*)
  591. Polygons Formed by Perpendicular Bisectors (*)
  592. Poncelet's Porism (*)
  593. Poncelet's Porism in Ellipses (*)
  594. Poncelet's Theorem (*)
  595. Ponzo Illusion (*)
  596. Ponzo Illusion II (*)
  597. Possible or Impossible? (*)
  598. Power of a point with respect to a circle
  599. Problem 4, 1975 USA Math Olympiad and the Radical Axis (*)
  600. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles (*)
  601. Problem 4, 1975 USA Math Olympiad: Normals and Tangents (*)
  602. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch (*)
  603. Problem 4 from the IMO 2009
  604. Problem in an Equilateral Triangle (*)
  605. Problem in an Equilateral Triangle II (*)
  606. Problem in Three Squares (*)
  607. Problem of Equal Steps (*)
  608. Problem of Equal Steps II (*)
  609. Problem on a Pentagonal Star
  610. Problem with angle bisector from 1975 Canadian Mathematical Olympiad
  611. Problem with equal angles from 1987 IberoAmerican Mathematical Olympiad
  612. Projections on Internal and External Angle Bisectors (*)
  613. Projective Collinearity in a Quadrilateral (*)
  614. Projective infinity
  615. Properties of the Circle of Similitude (*)
  616. Property of 6-Parpolygon (*)
  617. Property of Angle Bisectors (*)
  618. Property of Angle Bisectors II (*)
  619. Property of Circumscribed Quadrilaterals (*)
  620. Property of Points Where In- and Excircles Touch a Triangle (*)
  621. Ptolemy's Theorem
  622. Ptolemy on Hinges
  623. Puzzles on graphs.
  624. Pythagoras' Theorem (87 proofs)
  625. Pythagoras and Vecten Break Japan's Isolation (*)
  626. Pythagorean Triples (*)
  627. Quadrature: A Child's Play
  628. Quadrilateral With Equal Opposite Sides And Angles (*)
  629. Quadrilaterals Formed by Perpendicular Bisectors (*)
  630. Quest for Paragon (*)
  631. Radical Axis and Center, an Application (*)
  632. Radical Axis of Circles Inscribed in a Circular Segment (*)
  633. Radical Center (*)
  634. Radius and Construction of a Mixtilinear Circle (*)
  635. Radius of a Circle by Paper Folding (*)
  636. Random Dot Stereograms (*)
  637. Ratios and Orthogonality in a Triangle - Problem from the 2006 Romania MO
  638. Rational Points On a Circle
  639. Reflection In Line (*)
  640. Reflections in Ellipse (*)
  641. Reflections of a Line Through the Orthocenter (*)
  642. Reflections of a Point on the Circumcircle (*)
  643. Reflections of the Circumcenter
  644. Reflections of the Orthocenter (*)
  645. Reflections of the Orthocenter II
  646. Regular Polyhedra - Platonic Solids
  647. Other Solids
  648. Regular polyhedra and Euler's Theorem
  649. Relations in a Cyclic Polygon
  650. Remarkable Line in a Quadrilateral (*)
  651. Remarkable Line in Cyclic Quadrilateral (*)
  652. Rendezvous (*)
  653. Reverse Spoke Illusion (*)
  654. Revolving Circles Illusion (*)
  655. Rhombus in a Cyclic Quadrilateral (*)
  656. Riemann Sphere and Möbius Transformation
  657. Rotating Top Illusion (*)
  658. Rotating Gears (*)
  659. Rotation Transform (*)
  660. Rouse Ball's Fallacy (*)
  661. Rusty Compass Construction of Equilateral Triangle (*)
  662. Salinon: From Archimedes' Book of Lemmas (*)
  663. Sam Loyd's Geometric Puzzle
  664. Sam Loyd's Son's Dissection (*)
  665. Sangaku: Reflections and the Phenomenon
  666. Sangaku: Critique of My View and a Response
  667. Sangaku and The Egyptian Triangle
  668. Sangaku in a Square
  669. Sangaku Iterations, Is it Wasan? (*)
  670. Sangaku with 8 Circles
  671. Sangaku with Angle between a Tangent and a Chord (*)
  672. Sangaku with Quadratic Optimization
  673. Sangaku with Three Mixtilinear Circles (*)
  674. Sangaku with Versines (*)
  675. Sangakus with a Mixtilinear Circle (*)
  676. Scintillating Grid Illusion (*)
  677. Secant Angles (*)
  678. Secant Angles II (*)
  679. Secant, Normal, Tangent (*)
  680. Secant, Tangents and Orthogonality (*)
  681. See-Saw Lemma (*)
  682. Segment Trisection (*)
  683. Separating point sets with a circle
  684. Sequences of Touching Circles
  685. Seven Circles Theorem (*)
  686. Shaggy Dog Theorem
  687. Shapes in a lattice
  688. Shapes of constant width (*)
  689. Shearing and Translation in Pythagorean Pants (*)
  690. Shearing a Polygon into a Triangle of Equal Area (*)
  691. Shearing Butterflies in Quadrilaterals (*)
  692. Shepard's Parallelogram Illusion (*)
  693. Shifting Cube Illusion (*)
  694. Shimmer Illusion (*)
  695. Short Construction of the Geometric Mean
  696. Shortest Fence in a Quarter-Circle Pasture
  697. Shredding the torus
  698. Sierpinski Gasket By Common Trema Removal (*)
  699. Sierpinski Gasket Via Chaos Game (*)
  700. Sierpinski Gasket by Trema Removal (*)
  701. Similar Triangles And More
  702. Similar Triangles on Sides of a Quadrilateral (*)
  703. Similar Triangles on Sides and Diagonals of a Quadrilateral (*)
  704. Simple dissection of an acute triangle
  705. Simple Quadrilaterals Tessellate the Plane (*)
  706. Simson Line (*)
  707. Simsons and 9-Point Circles in Cyclic Quadrilateral (*)
  708. Simultaneous Contrast Illusion (*)
  709. Simultaneous Contrast Illusion II (*)
  710. Simultaneous Contrast: Koffka's Illusion (*)
  711. Simultaneous Contrast: White's Illusion (*)
  712. Simultaneous Contrast: White's Illusion, V-Variant (*)
  713. Simultaneous Generalization of the Theorems of Ceva and Menelaus
  714. Sine and Cosine of 15 Degrees Angle
  715. Sine, Cosine, and Ptolemy's Theorem
  716. Six Circles Theorem (Bui Quang Tuan) (*)
  717. Six Circles Theorem (Elkies) (*)
  718. Six Circles Theorem (Evelyn) (*)
  719. Six Concyclic Points (*)
  720. Six Concyclic Points II (*)
  721. Six Incircles in an Equilateral Triangle (*)
  722. Six Point Circle (*)
  723. SixPoints, ThreeLines (*)
  724. Soddy Circles and David Eppstein's Centers (*)
  725. Some Properties of Napoleon's Configuration (*)
  726. Sperner's Lemma (*)
  727. Spiral Similarity (*)
  728. Square and Circle in a Gothic Cupola
  729. Square From Nowhere (*)
  730. Square in a Circle Inscribed in a Square (*)
  731. Square in a Right Triangle (*)
  732. Square in Chair (*)
  733. Square Inscribed in a Triangle (*)
  734. Square Inscribed in a Triangle II (*)
  735. Squares in Circles (*)
  736. Squares Inscribed In a Triangle III (*)
  737. Squares on Sides of a Quadrilateral (*)
  738. Square out of a Quadrilateral
  739. Squaring a circle
  740. Squaring a Rectangle (*)
  741. Squinting Eyes Theorem (*)
  742. SSA
  743. SSS
  744. Staircase Illusion (*)
  745. Star of David (*)
  746. Steiner's Sangaku
  747. Steiner's Theorem
  748. Stereographic Projection (*)
  749. Stereographic Projection and Inversion
  750. Stereographic Projection and Radical Axes
  751. Stewart's theorem
  752. Straight Edge Only Construction of Polar
  753. Structural Constellation (*)
  754. Sum of Squares in Equilateral Triangle
  755. Sum of Squares of Distances to Vertices
  756. Symmedians
  757. Symmetries in a Triangle
  758. Tangency Point of Two Circles (*)
  759. Tangent and Secant (*)
  760. Tangent as a Radical Axis (*)
  761. Tangent Circles and an Isosceles Triangle (*)
  762. Tangent Circles in a Parallelogram (*)
  763. Tangent in Concurrency (*)
  764. Tangent Lines and Circles in Convex Quadrilateral (*)
  765. Tangent to two circles
  766. Tangents and Diagonals in Cyclic Quadrilateral (*)
  767. Tangents, Perpendiculars and Geometric Mean (*)
  768. Tarski-Banach Decompositions
  769. Taylor circle (*)
  770. Tesseract (*)
  771. The 80-80-20 Triangle
  772. The Chaos Game: Address Space vs IFS (*)
  773. The only rational triangle
  774. Thébault's Problem I (*)
  775. Thébault's Problem II (*)
  776. Thébault's Problem III (*)
  777. Thébault's Problem III, Proof (*)
  778. Thébault's Problem IV (*)
  779. Theorem of Complete Quadrilateral (*)
  780. Theorems of Ceva and Menelaus, an Illustrated Generalization (*)
  781. Three Circles and Common Tangents
  782. Three Circles through the Incenter
  783. Three Concurrent Circles (*)
  784. Three Congruent Circles by Reflection (*)
  785. Three Congruent Circles by Reflection II (*)
  786. Three Congruent Circles by Reflection III (*)
  787. Three Equal Circles in a Semicircle
  788. Three Incircles In a Right Triangle (*)
  789. Three Incircles in a Triangle (*)
  790. Three Isosceles Triangles (*)
  791. Three Orthogonal Circles Through Three Given Points (*)
  792. Three Parallels in a Triangle (*)
  793. Three Points on a Parabola (*)
  794. Three Pyramids are Better Than Two (*)
  795. Three regions sharing a common boundary
  796. Three Similar Polygons (*)
  797. Three Similar Triangles (*)
  798. Three Similar Triangles II (*)
  799. Three Squares and Two Ellipses (*)
  800. Three Tangent Circles (*)
  801. Three Tangents, Three Chords in Ellipse (*)
  802. Three Tangent Circles Sangaku
  803. Three Tangents Theorem (*)
  804. Three Tangents, Three Secants (*)
  805. Three Touching Circles (*)
  806. Toothpick Construction of a Square (*)
  807. Topological preliminaries
  808. Touching Circles (*)
  809. Touching Circles and Concurrency (*)
  810. Touching Circles with Given Centers II (*)
  811. Train wheel quandary
  812. Transitivity in Action (*)
  813. Translated Triangles (*)
  814. Translation Transform (*)
  815. Triangle ABC is right iff sin²A + sin²B + sin²C = 2
  816. Triangle Areas in a Parallelogram (*)
  817. Triangle Areas in a Parallelogram II (*)
  818. Triangle Classification (*)
  819. Triangle constructions and facts (*)
  820. Triangle Sliding on Two Straight Lines (*)
  821. Triangles, Squares and Areas from Temple Geometry (*)
  822. Triangles with Equal Area (*)
  823. Triangular Billiard (*)
  824. Triangulating Squares
  825. Trigonometric Form of Ceva's Theorem (*)
  826. Trigonometric Functions (*)
  827. Trigonometric Proof of the Pythagorean Theorem
  828. Trigonometry by Paper Folding
  829. Trisectable angle that is not constructible
  830. Tucker Circles (*)
  831. Tucker Circles Through Homothety (*)
  832. Twelve Matchsticks Area Puzzle
  833. Twisted Cord Illusion (*)
  834. Two Altitudes, One Midpoint
  835. Two Arbelos, Two Chains
  836. Two Circles and One More (*)
  837. Two Circles and Two More (*)
  838. Two Circles in a Parallelogram (*)
  839. Two Circles Inscribed in a Parallelogram (*)
  840. Two Circles in a Square (*)
  841. Two Circles in a Square II (*)
  842. Two Circles in an Angle (*)
  843. Two Circles on Angle Bisector (*)
  844. Two Circles on a Side of a Triangle (*)
  845. Two Circumcircles And Two Pairs of Parallels (*)
  846. Two Circumcircles in Triangle (*)
  847. Two coins puzzle
  848. Two Congruent Circles by Reflection (*)
  849. Two Color Coloring of the Plane (*)
  850. Two Equilateral Triangles (*)
  851. Two Homotheties in a Parallelogram (*)
  852. Two Intersecting Circles (*)
  853. Two Lines - Two Circles (*)
  854. Two Parallels in a Triangle and One More (*)
  855. Two Perpendiculars From a Point to a Line (*)
  856. Two Quadrilaterals (*)
  857. Two Quadruplets of Tangent Circles (*)
  858. Two Sangaku with Equal Incircles
  859. Two Simsons in a Triangle (*)
  860. Two Squares and Another Square (*)
  861. Two Symmetric Triangles Are Directly Decomposable (*)
  862. Two Touching Circles (*)
  863. Two Triangles With Common Base and Altitude (*)
  864. Two Triples of Concurrent Circles (*)
  865. Two Triples of Similar Triangles (*)
  866. Unfolding Spirals Illusion (*)
  867. Urquhart's Theorem
  868. USA 1975 Problem 4
  869. Use of Homogeneous Coordinates for Geometrical Calculations
  870. Van Obel Theorem, Determinants, and Barycentric Coordinates
  871. Van Schooten's and Pompeiu's Theorems (*)
  872. Van Schooten's Locus Problem (*)
  873. Variations on the Theme of Tremas (*)
  874. Variations on the Theme of Tremas II (*)
  875. Varignon Parallelogram (*)
  876. Various Geometries
  877. Vecten's Collinearity (*)
  878. Vecten's Mesh (*)
  879. Vector Addition and Subtraction [Flash]
  880. Vector product
  881. Versatile Theorem (*)
  882. Viewing a Statue
  883. Viviani's Theorem (*)
  884. Viviani's Theorem II (*)
  885. Wallace-Bolyai-Gerwien Theorem
  886. Wallace's Theorem (*)
  887. Weird curves bound normal areas
  888. Weitzenböck Inequality (*)
  889. What does one see inside a spherical mirror? (*)
  890. What Is Analytic Geometry?
  891. What Is Angle?
  892. What Is Ellipse?
  893. What Is Line?
  894. What Is Point?
  895. When a Triangle is Equilateral (*)
  896. When index equals content (*)
  897. Why a Diameter Is the Longest Chord?
  898. Wittenbauer's Parallelogram (*)
  899. Wundt Illusion (*)
  900. Wundt Block Illusion (*)
  901. Y. Sawayama's Lemma (*)
  902. Y. Sawayama's Theorem
  903. Zaslavsky's theorem (*)
  904. Zöllner's Illusion (*)

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Copyright © 1996-2010 Alexander Bogomolny

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