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 Quadrilateral - a Coffin Problem
6. 3-4-5 Triangle by a Kid
7. 4 Travelers problem
8. 6 to 9 Point Circle [Java]
9. 60^o Breeds 90^o [Java]
10. 7 = 2 + 5 Sangaku
11. 9-point Circle as a Locus of Concurrency [Java]
12. A better solution to a difficult sangaku problem
13. A Case of Similarity [Java]
14. A Chain of Touching Circles in a Polygon (a la Bui Quang Tuan) [Java]
15. A Chain of Touching Circles in a Polygon (à la Evelyn) [Java]
16. A Characterization of the Euler Line [Java]
17. A Circle in an Isosceles triangle from the 2006 Itish MO
18. A Circle Related to Incenter and Circumcenter [Java]
19. A Circle Rolling in an Equilateral Triangle [Java]
20. A Circle With Two Centers [Java]
21. A Corrected Version of the Calculus Proof of the Pythagorean Theorem
22. A Geometric Limit
23. A Hard but Important Sangaku
24. A Hinged Realization of a Plane Tessellation [Java]
25. A Lemma of Equal Areas [Java]
26. A Line in a Square Grid [Java]
27. A Line in Triangle Through the Circumcenter [Java]
28. A Multiplicative Identity of Areas in a Triangle [Java]
29. A Parallelogram in Triangle [Java]
30. A Possibly First Proof of the Concurrence of Altitudes
31. A Problem In an Isosceles Right Triangle [Java]
32. A Problem on an Icy Cone
33. A problem with equilateral triangles [Java]
34. A Proof Perigal and All Others After Him Missed
35. Properties of the Figures in Euclid I.47
36. A Property of Equiangular Polygons [Java]
37. A Property of Rhombi [Java]
38. A Property of the Line IO [Java]
39. A Property of the Line IO: A Bundled Statement and Proof [Java]
40. Property of the Line IO: a proof from the Book [Java]
41. A Relation in Triangle
42. A Restored Sangaku
43. A Sangaku Follow-Up on an Archimedes' Lemma [Java]
44. A Sangaku: Two Unrelated Circles [Java]
45. A Sangaku with an Egyptian Attachment [Java]
46. A Sangaku with Many Circles and Some [Java]
47. Semi-regular Tessellation on Hinges A [Java]
48. Semi-regular Tessellation on Hinges B [Java]
    • Semi-regular Tessellation on Hinges C {java]
49. A Stronger Triangle Inequality [Java]
50. A Sushi Morsel
51. A Three Pegs Question [Java]
52. A Triangle of Antreas Hatzipolakis [Java]
53. A Trigonometric Identity in a Right Triangle from the 1960 IMO
54. A Trigonometric Solution to a Difficult Sangaku Problem
55. An Area Inequality
56. About a Line and a Triangle [Java]
57. Abul Wafa al-Buzjani's Proof of the Pythagorean Theorem [Java]
58. Acute Angle Bisectors in a Right Triangle [Java]
59. Adams' Circle [Java]
60. Addition and Subtraction Formulas for Sine and Cosine
61. Addition and Subtraction Formulas for Sine and Cosine II
62. Algebraic proof of the theorem of butterflies in quadrilaterals
63. All Triangles Are Isosceles [Java]
64. Altitudes [Java]
65. Altitudes and the Euler Line [Java]
66. Altitudes and the Power of a Point [Java]
67. An Extension of van Schooten's Theorem
68. An Identity in a Regular Heptagon
69. An Isoperimetric Theorem [Java]
70. An Old Japanese Theorem
71. Angle: An Illustrated Classification [Java]
72. Angle Bisector in Equilateral Trapezoid
73. Angle Bisector in Parallelogram [Java]
    • IMO 2007, Problem 2, Second solution
74. Angle Bisectors In Rectangle [Java]
75. Angle Bisector in Touching Circles [Java]
76. Angle Bisectors [Java]
77. Angle Bisectors and Perpendiculars in a Quadrilateral [Java]
78. Angle Bisectors in Ellipse [Java]
79. Angle Bisectors in Ellipse II [Java]
80. Angle Bisectors in a Quadrilateral [Java]
81. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise [Java]
82. Angle Bisectors On Circumcircle [Java]
83. Angle Subtended by a Diameter [Java]
84. Angle Preservation Property [Java]
85. Angle Trisection, the Impossibility of Thereof
86. Angle Trisection by Archimedes [Java]
87. Angle Trisection by Hippocrates [Java]
88. Angle Trisection by Paul Vjecsner [Java]
89. Angle Trisectors on Circumcircle [Java]
90. Angle Trisection by Paper Folding
    • An Interesting Example of Angle Trisection by Paperfolding
91. Angles in a Cube I [Java]
92. Angles in a Cube II [Java]
93. Angles in a Cube III [Java]
94. Angles in Triangle Add to 180^o
95. Angles Inscribed in an Absent Circle [Java]
96. Anticomplementary Triangle Surprise [Java]
97. Antiparallel [Java]
98. Antiparallel via Three Reflections [Java]
99. APMO 1995 Problem 3
100. APMO 1998, Problem 4
101. APMO 2002, Problem 3
102. APMO 2003, Problem 2
103. APMO 2004, Problem 2
104. Apollonian Circles Theorem [Java]
105. Apollonian Gasket [Java]
106. Apollonius Problem [Java]
107. Approximate Construction of Regular Pentagon by A. Dürer
108. Arbelos -- the Shoemaker's Knife [Java]
     • Another Pair of Twins in an Arbelos [Java]
     • Archimedes' Quadruplets [Java]
     • Archimedes' Twin Circles and a Brother [Java]
     • Area of an Arbelos [Java]
     • Bisection of Yin and Yang
     • Chain of Circles on a Chord [Java]
     • Chain of Inscribed Circles [Java]
     • Chromatic Number of the Plane
     • Concurrency in Arbelos [Java]
     • Concyclic Points in Arbelos [Java]
     • Rectangle in Arbelos [Java]
     • Squares in Arbelos [Java]
     • Twin Segments in Arbelos [Java]
109. Archimedes' Book of Lemmas
     • Book of Lemmas: Proposition 1 [Java]
     • Book of Lemmas: Proposition 2 [Java]
     • Book of Lemmas: Proposition 3 [Java]
     • Book of Lemmas: Proposition 5 [Java]
     • Book of Lemmas: Proposition 6 [Java]
     • Book of Lemmas: Proposition 7
     • Book of Lemmas: Proposition 9 [Java]
     • Book of Lemmas: Proposition 10 [Java]
     • Book of Lemmas: Proposition 11
     • Book of Lemmas: Proposition 12 [Java]
     • Book of Lemmas: Proposition 13 [Java]
     • Book of Lemmas: Proposition 15
110. Archimedes' Law of the Lever [Java]
111. Archimedes' Method
112. Archimedes' Twins in the Edo Period
113. Area of a Bicentric Quadrilateral
114. Area of a Circle by Leonardo da Vinci [Java]
115. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
116. Area of Median Triangle [Java]
117. Area of Parallelogram [Java]
118. Area of the Union of Two Squares [Java]
119. Area of Triangle [Java]
120. Areas and Centroid in a Triangle [Java]
121. Areas In Circle
122. Equal Areas in Regular 2n-gons [Java]
123. Areas in Square by Dissection [Java]
124. Arithmetic-Geometric Mean Inequality
125. Arithmetic Mean Sangaku [Java]
126. Arithmetic Mean Sangaku II [Java]
127. Around the Incircle [Java]
128. Around the Incircle in a Polygon [Java]
129. Artisans vs Mathematicians
130. Asian Pacific Kite [Java]
131. Assigning Numbers to Points in the Plane
132. Assimilation Illusion [Java]
133. Asymmetric Propeller [Java]
134. Asymmetric Propeller by Plane Tiling [Java]
135. Axis of Similitude of Three Circles
136. Barbier, The Theorem of [Java]
137. Barycentric coordinates
     • Barycentric coordinates and Geometric Probability
     • Barycentric Coordinates: a Tool [Java]
138. Bender: A Visual Illusion [Java]
139. Between Major and Minor Circles [Java]
140. Bevan's Point and Theorem [Java]
141. Bimedians in a Regular Tetrahedron [Java]
142. Bisectal Circle [Java]
143. Bisecting arcs
144. Bisecting a shape
145. Bisector of an imaginary angle may be real
146. Bottema Shatters Japan's Seclusion [Java]
147. Bottema's theorem [Java]
     • An Elementary Proof of Bottema's Theorem
     • On Bottema's Shoulders [Java]
     • On Bottema's Shoulders II [Java]
148. Bottles in a Wine Rack [Java]
     • Bottles in a Slanted Rack [Java]
     • More Bottles in a Wine Rack [Java]
     • Bottles in a Wine Rack: Proofs and Generalizations [Java]
149. Bounded Distance
150. Brahmagupta's Formula and Theorem
151. Brahmagupta's Theorem [Java]
152. Brahmagupta-Mahavira Identities
153. Brianchon in Ellipse [Java]
154. Brianchon's theorem [Java]
155. Brick Wall Illusion [Java]
156. Bride's Chair [Java]
157. Brief introduction to Rational Trigonometry
158. Broken Chord Theorem [Java]
     • A Proof by Paper Folding [Java]
     • Broken Chord Theorem: Patruno's proof [Java]
     • Broken Chord Theorem: S. Anderson's proof [Java]
     • The Broken Chord Theorem: proof by Bui Quang Tuan
159. Building a Bridge [Java]
160. Building Bridges [Java]
161. Bulging lines illusion [Java]
162. Butterfly Theorem
     • A Better Butterfly Theorem
     • The Lepidoptera of the Circles [Java]
     • The Lepidoptera of the Quadrilateral
       • The Lepidoptera of the Quadrilateral II
     • Butterflies in Quadrilaterals and Elsewhere
     • Pinning Butterfly on Radical Axes [Java]
     • Two Butterflies Theorem [Java]
     • Two Butterflies Theorem II [Java]
     • Two Butterflies Theorem III [Java]
     • Butterfly in Ellipse [Java]
     • Butterflies in Hyperbola
     • The Plain Butterfly Theorem [Java]
     • Butterfly theorem, a Projective Proof
163. Cabart's Collinearity
164. Cantor Set and Function
     • Cantor sets [Java]
     • C₀ - C₀ = [-1, 1]
     • Difference of two Cantor sets, II
165. Cantor's Theorem [Java]
166. Carnot's Theorem
167. Carnot's Theorem
168. Carnot's Theorem (Generalization of Wallace's theorem) [Java]
169. Carpet With a Hole [Java]
170. Carpets Theorem [Java]
     • Carpets in a Parallelogram [Java]
     • Carpets in a Quadrilateral [Java]
     • Carpets in a Quadrilateral II [Java]
     • Carpets Theorem With Parallelograms [Java]
     • Two Rectangles in a Rectangle [Java]
     • Carpets in Hexagon
     • Round Carpets
171. Casey's Theorem [Java]
     • Three Points Casey's Theorem [Java]
172. centers of similitude
173. Centroid, a Characteristic Property Of
174. Centroids in a Polygon [Java]
175. Chasing Angles in Pascal's Hexagon [Java]
176. Circle, Isosceles Triangle and Fixed Point [Java]
177. Circle Centers on Radical Axes [Java]
178. Circle Chains on Napoleon Triangles [Java]
179. Circle in a Square Inscribed in a Circle [Java]
180. Circles on the sides of a triangle from the 2005 Austrian MO
181. Circles Tangent to Circumcircle [Java]
182. Circles with Tangent Diameters [Java]
183. Ceva and Menelaus Meet on the Roads
184. Ceva in Circumscribed Quadrilateral
185. Ceva's Theorem
186. Ceva's Theorem and Fibonacci Bamboozlement [Java]
     • A Faulty Dissection [Java]
187. Cevian Cradle [Java]
188. Cevian Nest [Java]
189. Cevian Triangle [Java]
190. Chain of Inscribed Circles [Java]
191. Chaos, Emergence of [Java]
192. Chasles' Theorem
     • Chasles' Theorem, a Proof [Java]
     • Chasles' Theorem, a Simple Proof
193. Cherchez le quadrilatère cyclique [Java]
194. Cherchez le quadrilatère cyclique II [Java]
195. Chords, Concurrency and Orthic Triangle [Java]
196. Chvatal's Art Gallery Theorem [Java]
197. Circle and Hyperbola as Lighthouse Curves
198. Circle and Hyperbola as Lighthouse Curves II
199. Circle Crosses Triangle [Java]
200. Circle Concurrency and Spiral Similarity [Java]
201. Circle Division by Euclid and Leonardo [Java]
202. Circle of Similitude [Java]
203. Circle through the Circumcenter [Java]
204. Circle through the Incenter [Java]
205. Circle through the Incenter And Antiparallels [Java]
206. Circle-Stacking Theorem [Java]
207. Circles And Parallels [Java]
208. Circles and Semicircles in Rectangle [Java]
209. Circles Cover a Quadrilateral [Java]
210. Circles in a Circular Segment
211. Circles in a Regular Polygon [Java]
212. Circles in Morley's Triangles [Java]
213. Circles Lined on the Legs of a Right Triangle [Java]
214. Circles On Cevians [Java]
215. Circles through the Orthocenter [Java]
216. Circles with Equal Collinear Chords [Java]
217. Circular Poggendorff Illusion [Java]
218. Circle Inscribed in a Circular Segment [Java]
219. Circumcenter on Angle Bisector [Java]
220. Circumcevian Triangle [Java]
221. Classification of Quadrilaterals [Java]
222. Cleavance Center [Java]
223. Cleaver [Java]
224. Clifford Algebras
225. Clifford's Chain [Java]
226. Coaxal Circles on Perpendicular Bisector [Java]
227. Coaxal Circles Theorem [Java]
228. Collage Theorem [Java]
229. Collinearity in Bicentric Quadrilaterals [Java]
230. Color Cycling on the Mandelbrot Set [Java]
231. Coloring Plane with Three Colors
232. Coloring Points in the Plane
233. Common Chord and a Tangent [Java]
234. Common Tangents to Two Circles I [Java]
235. Common Tangents to Two Circles II [Java]
236. Complete Quadrilateral [Java]
237. Complex Numbers and Geometry
238. Concurrency in the Intouch Triangle [Java]
239. Concyclic Circumcenters: A Dynamic View [Java]
     • Concyclic Circumcenters: A Sequel [Java]
240. Concyclic Points in a Triangle
241. Concyclic Points in Bride's Chair [Java]
242. Concyclic Points in Equilateral Bumps [Java]
243. Conic Sections
244. Conic Sections as Loci of Points [Java]
245. Conjugate Diameters in Ellipse [Java]
246. Consecutive Isosceles Decomposition [Java]
247. Constant Chord
248. Constructing a triangle from its angle bisectors is in general impossible
249. Construction of a Cyclic Quadrilateral
250. Construction of a Square from the 2008 Austrian MO
251. Construction of a Triangle from the Circumcenter, Orthocenter and Incenter
252. Construction of a Triangle from Two Vertices and the Centroid
253. Construction of Conics from Pascal's Theorem [Java]
254. Construction of Regular Pentagon by H. W. Richmond
255. Construction of the Angle Bisector [Java]
256. Construction of the Perpendicular Bisector [Java]
257. Construction problem [Java]
258. Constructions Related To An Inaccessible Point
259. Construction with the Compass Only
260. Continuous Line Illusion [Java]
261. Convex Polygon Is the Intersection of Half Planes [Java]
262. Copernicus' Theorem [Java]
     • Proof of Copernicus' Theorem [Java]
263. Construction with Optimization in Angle - Problem the 1979 USA Mathematical Olympiad
264. cos(36°) = (1 + √5)/4
265. Cosine Law
     • Laws of Sines and Cosines [Java]
     • The Illustrated Law of Cosines
     • The Law of Cosines: a proof without words [Java]
     • The Law of Cosines: An Unfolded Variant [Java]
     • The Law of Cosines (Independent of the Pythagorean Theorem)
266. Counting Triangles and Segments in a Polygon [Java]
267. Cross Points in a Polygon [Java]
268. Cross-Ratio
269. Crossed-Lines Construction of Shapes of Constant Width [Java]
270. Crossing Number of a Graph
271. Cube Counting Ambiguity [Java]
272. Curry's Paradox [Java]
273. Cut the Cone [Java]
274. Cut the Cube [Java]
275. Cut the Cylinder [Java]
276. Cutting Squares
277. Cutting Triangles off Regular Hexagon
     • A Property of Centrally Symmetric Hexagons
278. Cyclic Hexagon [Java]
279. Cyclic Quadrilateral, Concurrent Circles and Collinear Points [Java]
280. Cycloids [Java]
281. Dancing Rectangles Model Auxetic Behavior [Java]
282. Dancing Squares or a Hinged Plane Tessellation [Java]
283. David Richeson's Extention of an Old Japanses Theorem [Java]
284. Day's Sine Illusion [Java]
285. Decomposition of Isosceles Triangles into 4 Isosecles Triangles
286. Delboeuf Illusion [Java]
287. Delian Problem Solved [Java]
288. Desargues in the Bride's Chair (with Pythagoras)
289. Desargues' Theorem [Java]
290. Determination of
291. Diagonal Count [Java]
292. Diagonals in a Cyclic Quadrilateral
293. Diameters and Chords [Java]
294. Directly Similar Figures [Java]
295. Disappearing Lines puzzle [Java]
296. Discovery of Duane DeTemple [Java]
297. Dissection of a 10×13 Rectangle into Two Chessboards
298. Dissection of Cyclic Quadrilateral [Java]
299. Dissection of Triangle into Rhombus [Java]
300. Dissection of Triangle into Rhombus by Hubert Shutrick [Java]
301. Distance
302. Distance between the Orthocenter and Circumcenter
303. Distance Formula
304. Distorted Lines Illusion [Java]
305. Divergent Spirals Illusion [Java]
306. Divide a Circle into N Parts of Equal Area
307. Divide a segment into n equal parts
     • Dividing a Segment by Paper Folding
     • Dividing a Segment into N parts: Besteman's Construction [Java]
     • Dividing a Segment into N parts: Besteman's Construction II [Java]
     • Dividing a Segment into N parts: GLaD Construction [Java]
     • Dividing a Segment into N parts: SaRD Construction [Java]
     • Al-Nayrizi's Construction [Java]
     • Euclid's Segment Division [Java]
     • Similar Right Triangles [Java]
308. Dividing Evenly a Quadrilateral [Java]
309. Dividing Evenly a Quadrilateral II [Java]
310. Double Generation Theorem [Java]
311. Drawing circles [Java]
312. Droz-Farny Circles [Java]
313. Droz-Farny Line Theorem [Java]
314. Duplication of the Cube
315. Dynamic Construction of Ellipse and Other Curves [Java]
316. E. W. Dijkstra's Proof of the Pythagorean Theorem [Java]
317. Easy Construction of Bicentric Quadrilateral [Java]
318. Easy Construction of Bicentric Quadrilateral II [Java]
319. Egyptian Triangle By Paper Folding
320. Eight Equal Tangents [Java]
321. Eight Point Circle [Java]
322. Ellipse in Arbelos [Java]
323. Ellipse Between Two Circles [Java]
324. Equal Angles in Two Circles [Java]
325. Equal Angles in Two Circles II [Java]
326. Equal Areas in Equilateral Triangle [Java]
327. Equal Areas In Parallelogram à la Pythagoras [Java]
328. Equal Chords in Crossing Circles [Java]
329. Equal Circles, Medial Triangle and Orthocenter [Java]
330. Equal Incircles Theorem [Java]
     • Equal Incircle Theorem, Angela Drei's Proof
331. Equichordal points: Just Do It [Java]
332. Equidecomposition of a Rectangle and a Square [Java]
333. Equidecomposition of Two Rectangles [Java]
334. Equidecomposition of a Triangle and a Rectangle [Java]
335. Equidecomposition of a Triangle and a Rectangle II [Java]
336. Equidecomposition of Two Parallelograms [Java]
337. Equilateral and 3-4-5 Triangles [Java]
338. Equilateral Triangle on Parallel Lines [Java]
339. Equilateral Triangle, Straight Line and Tangent Circles [Java]
340. Equilateral Triangles and Incircles in a Square
341. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral [Java]
342. Equilateral Triangles On Sides of a Parallelogram [Java]
343. Equilateral Triangles On Sides of a Parallelogram II [Java]
344. Equilateral Triangles on Sides of a Quadrilateral [Java]
345. Equilic Quadrilateral I, a Flawed Proof [Java]
346. Equilic Quadrilateral I, A Variation [Java]
347. Equilic Quadrilateral II [Java]
348. Erdös-Mordell Inequality
349. Escher's Theorem [Java]
350. Euclid's Fifth Postulate
     • Similarity and the Parallel Postulate
351. Euclid's Elements Reference Page
352. Euclid I.43 [Java]
353. Euclid I.43 Extended [Java]
354. Euclidean Construction of Center of Ellipse [Java]
355. Euclidean Construction of Tangent to Ellipse
356. Euler Line Cuts Off Equilateral Triangle [Java]
357. Euler Line and 9-Point Circle [Java]
358. Euler's Formula and Poncelet Porism
359. Every Parallelogram Is a Rectangle
360. Existence of the Circumcenter [Java]
361. Existence of the Euler Line: An Elementary Proof [Java]
362. Experimentation with Dynamic Geometry Software: An Example [Java]
363. Exterior Angle Theorem - an appreciation
364. External Angle Bisector [Java]
365. Extouch Triangle in Poncelet Porism [Java]
366. Extremal Problem in a Circular Segment
367. Eyeball Theorem [Java]
368. Fagnano's problem [Java]
     • Fagnano's Problem, Third Solution [Java]
     • Fagnano's Problem, Izvolsky's Solution [Java]
     • Fagnano's Problem In Reverse [Java]
369. Faulty Symmetry
370. Fermat point and 9-point Centers [Java]
     • Fermat's Point and Generalizations [Java]
     • Fermat's Point [Java]
     • Fermat Points and Concurrent Euler Lines I [Java]
     • Fermat Points and Concurrent Euler Lines II [Java]
371. Feuerbach's Theorem: A Proof [Java]
372. Find a Common Chord of Given Length [Java]
373. Find the Angle! [Java]
374. Finding Circle Center By Ruler Alone [Java]
375. Five Concyclic Points [Java]
376. Five Incircles in a Square [Java]
377. Five Incircles Theorem [Java]
378. Fixed Point in a Kite - Problem from the 1993 Asian Pacific MO
379. Fixed Point in Isosceles and Equilateral Triangles
380. Fixed Point of Circles Orthogonal to the Given One [Java]
381. Fixed points from 2007 Irish Mathematical Olympiad
382. Fleury's Algorithm and Euler's Paths and Cycles [Java]
383. Following the Hilbert Curve [Java]
384. Foot of Altitude and Minimum Distance [Java]
385. Focal Definition of Ellipse [Java]
386. Focus and Directrix of Ellipse [Java]
387. For every object there is a distance at which it looks its best.
388. Ford's touching circles
389. Four Centroids and Parallels [Java]
390. Four Circles In a Triangle [Java]
391. Four Concurrent Lines in a Cyclic Quadrilateral [Java]
392. Four Construction Problems [Java]
393. Four Crossing Circles [Java]
394. Four Geometric Problems of Antiquity
395. Four Hinged Squares [Java]
396. Four Incident Circles [Java]
397. Four Incircles in Equilateral Triangle [Java]
398. Four 9-Point Circles in a Quadrilateral [Java]
399. Four Pedal Circles [Java]
400. Four Touching Circles [Java]
401. Four Touching Circles II [Java]
402. Four Touching Circles III [Java]
403. Four Triangles, One Circle [Java]
404. Four Turtles (or Bugs, or Dogs, etc.) [Java]
405. Fractal curves and dimension [Java]
406. Friendly Kiepert's Perspectors [Java]
407. From Equilateral Triangle and Square to Golden Ratio
408. From Foci to a Tangent in Ellipse [Java]
409. Fundamental Theorem of 3-Bar Motion [Java]
410. Fuss' Theorem [Java]
411. Generalizing a Romanian Olympiad Problem [Java]
412. Geometric Illustration of a Convergent Series
413. Geometric Mean Sangaku
414. Geometric Meaning of the Geometric Mean
415. Geometric Optimization from the Asian Pacific Mathematical Olympiad [Java]
416. Geometry, Algebra, and Illustrations
417. Gergonne and Medial Triangles Are Orthologic [Java]
418. Gergonne and Soddy Lines Are Perpendicular [Java]
419. Gergonne in Ellipse [Java]
420. Gergonne and Van Obel Theorems
421. Gion Shrine Problem
422. Given Parabola, Find Axis
423. Glide Reflection [Java]
424. Golden Ratio in Geometry
425. Golden Ratio and Midpoints
426. Golden Ratio in an Irregular Pentagon
     • Golden Ratio in an Irregular Pentagon, Construction I [Java]
     • Golden Ratio in an Irregular Pentagon, Construction II [Java]
     • Golden Window
427. Golomb's inductive proof of a tromino theorem [Java]
     • Tromino as a Rep-tile [Java]
     • Tiling Rectangles with L-Trominoes [Java]
428. Gothic Arc [Java]
429. Graph Theory [Java]
430. Grebe: from Ladies' Diary to Carroll's Pillow [Java]
431. Grebe's Theorem
432. Greg Markowsky's Problem for Parabola [Java]
433. Griffiths Points in Cyclic Quadrilateral [Java]
434. Half Turn, Reflection in Point [Java]
435. Halving a square
436. Harmonic Mean in Geometry
437. Harmonic Mean Sangaku [Java]
438. Harmonic Ratio
439. Harmonic Ratio in Complex Domain
440. Hart Circle [Java]
441. Hart's Inversor [Java]
442. Haruki's Lemma [Java]
443. Haruki's Theorem
444. Hausdorff Distance
445. Hearty Munching on Cardioids [Java]
446. Height/Width Interplay Illusion [Java]
447. Helly's Theorem
     • First Applications of Helly's Theorem
448. Hering Illusion [Java]
449. Hermann Grid Illusion [Java]
450. Heron's formula
451. Heron's Formula: a Proof [Java]
452. Heron's Problem [Java]
453. Hexagon Parallel to Medial Triangle [Java]
454. Hexagon Parallel to Orthic Triangle [Java]
455. Hexagons in a Square
456. Hinged Greek Cross Tessellation [Java]
457. Hinged Squares [Java]
458. Hippocrates' Squaring of a Lune [Java]
459. Hippocrates' Squaring of Lunes
460. Hjelmslev Theorem [Java]
461. Homologous Lines under Three Spiral Similarities [Java]
462. Homothety [Java]
463. Homothety between In- and Excircles [Java]
464. Hooper's Paradox [Java]
465. Hunting Right Angles [Java]
466. Ian McGee's Observation [Java]
467. IMO 1998 Problem 1
468. IMO 2004, Problem 5
469. IMO 2006, Problem 1
470. Impossible Fork [Java]
471. Impossible Frame [Java]
472. Importance of Occluders: Motion Binding [Java]
473. In the Wasan Spirit
474. Incenters in Cyclic Quadrilateral [Java]
475. Incidence in Feuerbach's Theorem [Java]
476. Infinitude of Primes - A Topological Proof
477. Inscribed and Central Angles in a Circle [Java]
478. Inscribed Angles [Java]
479. Inscriptible and Exscriptible Quadrilaterals [Java]
480. Inscriptible Quadrilateral: An Illustration [Java]
481. Internal Tangents to Three Circles [Java]
482. Intersecting Chords Theorem
483. Intersections of a Circle with the Four Quadrants [Java]
484. Intouch Triangle in Poncelet Porism [Java]
485. Invariance in Orthodiagonal Quadrilaterals [Java]
486. Invariant Area Sweep Implies Pythagorean Theorem [Java]
487. Inversion: Reflection in a Circle [Java]
     • Inversion [Java]
     • Inversion in the Incircle [Java]
     • Inversion Tool
     • Inversion with Negative Power [Java]
488. Involution on the Projective Line
489. Is Every Trapezoid Parallelogram? [Java]
490. Isosceles Tetrahedron
491. Isogonal and Isotomic Conjugalities [Java]
492. Isogonal Concurrency [Java]
493. Isogonal image of the circumcircle [Java]
494. Isoperimetric Theorem and Inequality
     • Isoperimetric Property of Equilateral Triangles
     • Isoperimetric Theorem for Rectangles
     • A Variant of the Isoperimetric Theorem
495. Isosceles on the Sides of a Triangle [Java]
496. Isotomy and Isogonality Hand-in-Hand [Java]
497. Iterations and the Mandelbrot Set [Java]
498. Iterations on a Circle Through Three Points [Java]
499. Iterations on Euler Lines [Java]
500. J. Casey's Additions to Euclid I.47
501. Jan Stevens' proof of E. W. Dijkstra's Generalization of the Pythagorean theorem
502. Joachimsthal's Notations
503. John Sharp's Paradox [Java]
504. Johnson Circles [Java]
505. Join Circles by Given Segment [Java]
506. Joseph Keech in Bride's Chair [Java]
507. Judd Illusion [Java]
508. Julia sets indexing [Java]
509. Kanizsa Triangle [Java]
510. Kiepert's Theorem [Java]
511. Knots [avi movie]
512. Knots on a Torus [Java]
513. Kürschak's Tile and Theorem [Java]
514. La Hire's Theorem [Java]
     • La Hire's Theorem [Java]
     • La Hire's Theorem in Ellipse [Java]
515. Langman's Paradox [Java]
516. Lean Napoleon's Triangles [Java]
517. Length of the diagonal of the unit square equals the square root of 2
518. Lennes' Polyhedron [Java]
519. Lepidoptera of the Circles [Java]
520. Line IO in Bicentric Quadrilaterals
521. Line through a Point in an Angle
522. Line, Circle, and Fixed Points [Java]
523. Loci of points in a cube
524. Locus of Points in a Given Ratio to Two Points [Java]
525. Longest segment
526. MacLaurin's Construction of Conics [Java]
527. Maimed Cake [Java]
528. Malfatti's Problem [Java]
529. Malfatti's Problem, Hart's Solution [Java]
530. Mascheroni Construction of a Regular Pentagon
531. Mathematical Droodles [Java]
532. Mathematics in Pizzeria [Java]
533. MathPro Logo [Java]
534. Maximal Properties of the Pythagorean Relation
535. Maximum Area Property of Equilateral Triangles
536. Maximum Area Property of the Medial Triangle
537. Maximum Perimeter Property of the Incircle {*}
538. Maximum Ratio of Similarity - Belarus Olympiad Problem
539. Maxwell's Theorem [Java]
     • Maxwell Theorem via the Center of Gravity
     • Projective Proof of Maxwell's Theorem
     • Projective Generalization of Maxwell's Theorem [Java]
540. May the Product of Planes Be a Sphere?
541. Mechanical Proofs of the Pythagorean Theorem
542. Median in Touching Circles [Java]
543. Medians
544. Medians in a Quadrilateral [Java]
545. Menelaus Theorem [Java]
     • Menelaus Theorem: Proofs Ugly and Elegant - A. Einstein's View [Java]
     • Menelaus From Ceva [Java]
     • Menelaus from 3D
546. Metric Relations in a Triangle
547. Midline in Similar Triangles [Java]
548. Midpoints and Orthogonality in Isosceles Triangles [Java]
549. Midpoints from Gergonne Triangle [Java]
550. Midpoints of the Lines Joining In- and Excenters [Java]
551. Mikhalev's Octahedrons
552. Minesweeper Theorem
553. Miquel's Point [Java]
554. Miquel's Point of a 4-line Via Spiral Similarity [Java]
555. Miquel's Theorem for Circles [Java]
556. Mirror on the Wall [Java]
557. Mirror property of the altitudes [Java]
558. Mirror Property of Altitudes via Pascal's Hexagram [Java]
559. Mixtilinear Circles and Concurrence [Java]
560. Moebius Strip [avi movie]
561. Möbius' Strip [Java]
562. Models and Metamathematics
563. Monge via Desargues [Java]
564. Monge via Desargues II [Java]
565. Morley Constellation [Java]
566. Morley's Miracle [Java]
     • Doodling and Miracles [Java]
     • Morley's Pursuit of Incidence [Java]
     • Lines, Circles and Beyond [Java]
     • Morley's Theorem, G. Zsolt Kiss' proof [Java]
     • Morley's Theorem, R. Smyth's Proof
     • Proof by H. D. Grossman
     • Proof by R. J. Webster
     • On Motivation and Understanding [Java]
     • Bankoff's Conundrum
     • B. Bollobás' proof
     • Morley's Redux and More [Java]
     • An Unexpected Variant
     • Morley's Theorem: Proof by B. Stonebridge and B. Millar
     • Morley's Theorem: Proof by H. Shutrick
567. Morphing [Java]
568. Müller-Lyer Illusion [Java]
569. Munching on Circles
570. Munching on Inscribed Angles
     • More On Inscribed Angles [Java]
571. My logo
572. Nagel Line [Java]
573. Nagel point [Java]
574. Nagel Point of the Medial Triangle [Java]
575. Nagel's Theorem [Java]
576. Napoleon's Theorem
     • Napoleon-Barlotti Theorem [Java]
     • Napoleon on Hinges [Java]
     • Napoleon's Relatives [Java]
     • Napoleon's Theorem, a generalization
     • Napoleon's Theorem by Plane Tesselation [Java]
     • Napoleon's Theorem by Transformation [Java]
     • Generalization of Napoleon's Theorem [Java]
     • Napoleon's Propeller [Java]
     • Napoleon's Theorem via Two Rotations [Java]
     • Napoleon's Theorem: A third proof with complex numbers [Java]
577. Necker Cube [Java]
578. Neuberg Sangaku [Java]
579. Newton's Construction of Conics [Java]
580. Newton's Theorem [Java]
581. Nine Point Circle [Java]
582. Nine Point Circle: an Elementary Proof [Java]
583. No Equilateral Triangles, Please [Java]
584. Nobbs' Points, Gergonne Line [Java]
585. Non-Eclidean Geometries
586. No-Pedal Collinearity [Java]
587. Occluded Wheel Illusion [Java]
588. Olof Hanner's Jigsaw Puzzle [Java]
589. On the Difference of Areas
590. One Sheet Hyperboloid [Java]
591. Optimization Problem in Acute Angle [Java]
592. Octagon in Parallelogram: Overlapping Regions [Java]
593. Octagon in Parallelogram [Java]
594. Optical Property of Ellipse [Java]
595. Orbison illusion [Java]
596. Orthocenter and Three Equal Circles [Java]
597. Orthocenters
598. Orthocenters of Two Triangles Sharing Circumcenter and Base [Java]
599. Orthocentric System From Rectangle [Java]
600. Orthodiagonal and Cyclic Quadrilaterals [Java]
601. Orthogonality in Isosceles Triangles [Java]
602. Orthologic Triangles in a Quadrilateral [Java]
603. Orthopole [Java]
604. Out of Pentagon Sangaku
605. Packing shapes in a plane
606. Pairs of Areas in Equilateral Triangle
607. Pairs of Homologous Lines under Spiral Similarities [Java]
608. Pairs of Incircles in a Quadrilateral [Java]
609. Paper Folding & Cutting Sangaku
610. Paper Folding Geometry
611. Paper strip activities [avi movies]
612. Paper strip activities, two strips
613. Pappus' Generalization of Euclid I.47 [Java]
614. Pappus' Theorem [Java]
     • Dual to Pappus'Theorem [Java]
615. Parabola [Java]
616. Parabolic Mirror, Illustration [Java]
617. Parabolic Mirror, Theory [Java]
618. Paragon [Java]
619. Parahexagon [Java]
620. Parallel Chords [Java]
621. Parallel Chords II [Java]
622. Parallel Chords in Ellipse [Java]
623. Parallel Chords in Crossing Circles [Java]
624. Parallel Lines in a Cyclic Quadrilateral [Java]
625. Parallel Lines in a Quadrilateral [Java]
626. Parallelogram and Similar Triangles [Java]
627. Parallelogram Illusion [Java]
628. Parallelogram Iterations [Java]
629. Parallelogram Law [Java]
630. Parallelogram Law: A PWW [Java]
631. Parallelogram Law: A Tessellation [Java]
632. Partition of point sets in a plane
633. Partitioning 3-Space with Circles
634. Partitioning a Circle
635. Pascal: Necessary and Sufficient [Java]
636. Pascal in a Cyclic Quadrilateral [Java]
637. Pascal Lines: Steiner and Kirkman Theorems [Java]
638. Pascal Lines: Steiner and Kirkman Theorems II [Java]
639. Pascal's theorem [Java]
     • Pascal's Theorem, Homogeneous Coordinates
     • Illustration of Pascal's Theorem [Java]
     • Projective Proof of Pascal's Theorem [Java]
640. Peacock's Tail Sangaku
641. Pedal Collinearities [Java]
642. Pedal Polygons [Java]
643. Pedal Triangle and Isogonal Conjugacy [Java]
644. Pedoe's Theorem [Java]
645. Pencils of Cubics
646. Pentagon, construction of
     • Regular Pentagon Construction by Y. Hirano
647. Pentagon in a Semicircle [Java]
648. Pentagon Proportions Sangaku
649. Periodic patterns in sin(N) [Java]
650. Perpendicular Bisectors in an Inscriptible Quadrilateral [Java]
651. Perpendicular Bisectors in an Inscriptible Quadrilateral II [Java]
652. Perpendicular lines in a cyclic quadrilateral from 1975 Canadian Mathematical Olympiad
653. Phantom Circle and Recaptured Symmetry
654. Pivot Theorem [Java]
655. Plane Filling Curves [Java]
     • Plane Filling Curves: One of Sierpinski's Curves [Java]
     • Plane Filling Curves: the Lebesgue Curve [Java]
     • All Peano Curves [Java]
656. Plane Isometries
657. Plato's Geometric Numbers: can you better Sam Loyd?
658. Peaucellier Linkage [Java]
659. Pick's Theorem [Java]
660. Poggendorff's Illusion
661. Point common to two similar rectangles [Java]
662. Point in a Square
663. Point on Bisector in Right Angle
664. Points Generated by the Nine Points
665. Polar Circle [Java]
666. Poles and Polars [Java]
     • Poles, Polars and Quadrilaterals [Java]
     • Pole and Polar with respect to a Triangle [Java]
667. Polygon in a Rectangle
668. Polygons: formality and intuition [Java]
669. Polygons Formed by Perpendicular Bisectors [Java]
670. Pompeiu's Theorem [Java]
671. Poncelet's Porism [Java]
672. Poncelet's Porism in Ellipses [Java]
673. Poncelet's Theorem [Java]
674. Ponzo Illusion [Java]
675. Ponzo Illusion II [Java]
676. Possible or Impossible? [Java]
677. Power of a point with respect to a circle
678. Problem 4, 1975 USA Math Olympiad and the Radical Axis [Java]
679. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles [Java]
680. Problem 4, 1975 USA Math Olympiad: Normals and Tangents [Java]
681. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch [Java]
682. Problem 4 from the IMO 2009
683. Problem in an Equilateral Triangle [Java]
684. Problem in an Equilateral Triangle II [Java]
685. Problem in Three Squares [Java]
686. Problem of Equal Steps [Java]
687. Problem of Equal Steps II [Java]
688. Problem on a Pentagonal Star
689. Problem with angle bisector from 1975 Canadian Mathematical Olympiad
690. Problem with equal angles from 1987 IberoAmerican Mathematical Olympiad
691. Projections on Internal and External Angle Bisectors [Java]
692. Projective Collinearity in a Quadrilateral [Java]
693. Projective infinity
694. Properties of the Circle of Similitude [Java]
695. Property of 6-Parpolygon [Java]
696. Property of Angle Bisectors [Java]
     • Angle Bisector Theorem
697. Property of Angle Bisectors II [Java]
698. Property of Circumscribed Quadrilaterals [Java]
699. Property of Points Where In- and Excircles Touch a Triangle [Java]
700. Proportions in Square
701. Ptolemy's Theorem
     • Ptolemy by Inversion
     • Ptolemy on Hinges
     • Ptolemy via Cross-Ratio
702. Puzzles on graphs.
703. Pythagoras' Theorem (91 proofs)
     • A Neglected Pythagorean-Like Formula
     • Huygens' proof
     • Pythagorean Theorem by Hinged Dissection
     • Pythagorean Theorem by Hinged Dissection I
     • Pythagorean Theorem by Hinged Dissection II
     • Pythagorean Theorem from the Shoelace Formula
     • Pythagorean Theorem via an Isosceles Triangle [Java]
     • False Proofs of the Pythagorean Theorem
     • Pythagoras' from Bottema's [Java]
     • Pythagoras' Theorem By Sheer Shearing [Java]
     • Pythagoras' Theorem By Similarity [Java]
     • Pythagoras' Theorem Solely By Sliding [Java]
     • Pythagorean Condition in An Isosceles Right Triangle [Java]
     • Pythagorean Theorem: J. Adams' Proof by Dissection [Java]
     • Pythagorean Theorem: Subtle Dangers of Visual Proof [Java]
     • Pythagorean Theorem By Plane Tessellation [Java]
     • Pythagorean Theorem, Generalizations of
     • Pythagorean Theorem in an Orthodiagonal Quadrilateral [Java]
     • Pythagorean Theorem is Equivalent to Parallel Postulate.
     • Pythagorean Theorem: Versluys' Proof by Dissection [Java]
     • Mechanical Proof of the Pythagorean Theorem
     • "Extra-geometric" proofs of the Pythagorean Theorem
     • D. Houston's proof: Is it trigonometric?
     • Pythagoras' from Broken Chords
     • Pythagoras' from the Star of David
     • Pythagorean Theorem: An Algebraic Proof with Three Trapezoids
704. Pythagoras and Vecten Break Japan's Isolation [Java]
705. Pythagorean Triples [Java]
706. Quadrature: A Child's Play
707. Quadrilateral With Equal Opposite Sides And Angles [Java]
708. Quadrilateral with 3 Equal Sides [Java]
709. Quadrilaterals Formed by Perpendicular Bisectors [Java]
710. Quest for Paragon [Java]
711. Radical Axis and Center, an Application [Java]
712. Radical Axis of Circles Inscribed in a Circular Segment [Java]
713. Radical Center [Java]
714. Radius and Construction of a Mixtilinear Circle [Java]
715. Radius of a Circle by Paper Folding [Java]
716. Random Dot Stereograms [Java]
717. Ratios and Orthogonality in a Triangle - Problem from the 2006 Romania MO
718. Rational Points On a Circle
719. Reflection In Line [Java]
720. Reflections in Ellipse [Java]
721. Reflections of a Line Through the Orthocenter [Java]
722. Reflections of a Point on the Circumcircle [Java]
723. Reflections of the Circumcenter
724. Reflections of the Orthocenter [Java]
725. Reflections of the Orthocenter II
726. Regular Polygons in a Triangular Grid [Java]
727. Regular Polyhedra - Platonic Solids
     • Tetrahedron [Java]
     • Octahedron [Java]
     • Cube [Java]
     • Icosahedron [Java]
     • Dodecahedron [Java]
     • Cube In Octahedron [Java]
     • Octahedron In Cube [Java]
     • Icosahedron In Cube [Java]
728. Other Solids
     • Pentagonal Prism
     • Square Pyramid
     • Triangular Prism
     • Great Stellated Dodecahedron
     • Triangulated Dinosaur
729. Regular Pentagon Inscribed in Circle by Paper Folding
730. Regular polyhedra and Euler's Theorem
731. Relations in a Cyclic Polygon
732. Remarkable Line in a Quadrilateral [Java]
733. Remarkable Line in Cyclic Quadrilateral [Java]
734. Rendezvous [Java]
735. Reverse Spoke Illusion [Java]
736. Revolving Circles Illusion [Java]
737. Rhombus in a Cyclic Quadrilateral [Java]
738. Riemann Sphere and Möbius Transformation
739. Right Triangles on Sides of a Square [Java]
740. Rotating Top Illusion [Java]
741. Rotating Gears [Java]
742. Rotation Transform [Java]
     • Product of Rotations [Java]
     • Rotations in Disguise [Java]
743. Rouse Ball's Fallacy [Java]
744. Rusty Compass Construction of Equilateral Triangle [Java]
745. Salinon: From Archimedes' Book of Lemmas [Java]
746. Sam Loyd's Geometric Puzzle
747. Sam Loyd's Son's Dissection [Java]
748. Sangaku: Reflections and the Phenomenon
749. Sangaku: Critique of My View and a Response
750. Sangaku and The Egyptian Triangle
751. Sangaku in a Square
752. Sangaku Iterations, Is it Wasan? [Java]
753. Sangaku with 8 Circles
754. Sangaku with Angle between a Tangent and a Chord [Java]
755. Sangaku with Quadratic Optimization
756. Sangaku with Three Mixtilinear Circles [Java]
757. Sangaku with Versines [Java]
758. Sangakus with a Mixtilinear Circle [Java]
759. Scintillating Grid Illusion [Java]
760. Schwarz Lantern [Java]
761. Secant Angles [Java]
762. Secant Angles II [Java]
763. Secant, Normal, Tangent [Java]
764. Secant, Tangents and Orthogonality [Java]
765. See-Saw Lemma [Java]
766. Segment Trisection [Java]
767. Segment Trisection Induced by Parallels to Medians [Java]
768. Separating point sets with a circle
769. Sequences of Touching Circles
770. Serendipitous Beauty [Java]
771. Seven Circles Theorem [Java]
772. Shaggy Dog Theorem
773. Shapes in a lattice
774. Shapes of constant width [Java]
     • Star Construction of Shapes of Constant Width [Java]
775. Shearing and Translation in Pythagorean Pants [Java]
776. Shearing a Polygon into a Triangle of Equal Area [Java]
777. Shearing Butterflies in Quadrilaterals [Java]
778. Shepard's Parallelogram Illusion [Java]
779. Shifting Cube Illusion [Java]
780. Shimmer Illusion [Java]
781. Short Construction of the Geometric Mean
782. Shortest Fence in a Quarter-Circle Pasture
783. Shredding the torus
784. Sierpinski Gasket By Common Trema Removal [Java]
785. Sierpinski Gasket Via Chaos Game [Java]
786. Sierpinski Gasket by Trema Removal [Java]
787. Similar Triangles And More
788. Similar Triangles on Sides of a Quadrilateral [Java]
789. Similar Triangles on Sides and Diagonals of a Quadrilateral [Java]
790. Simple dissection of an acute triangle
791. Simple Quadrilaterals Tessellate the Plane [Java]
792. Simson Line [Java]
     • Simson Line From Isogonal Perspective [Java]
     • Simson Line in Disguise [Java]
     • Simson Lines Of Diametrically Opposite Points [Java]
793. Simsons and 9-Point Circles in Cyclic Quadrilateral [Java]
794. Simultaneous Contrast Illusion [Java]
795. Simultaneous Contrast Illusion II [Java]
796. Simultaneous Contrast: Koffka's Illusion [Java]
797. Simultaneous Contrast: White's Illusion [Java]
798. Simultaneous Contrast: White's Illusion, V-Variant [Java]
799. Simultaneous Diameters in Concurrent Circles [Java]
     • Three Concurrent Circles and Simultaneous Diameters
800. Simultaneous Generalization of the Theorems of Ceva and Menelaus
801. Sine and Cosine of 15 Degrees Angle
802. Sine, Cosine, and Ptolemy's Theorem
803. Six Circles Theorem (Bui Quang Tuan) [Java]
804. Six Circles Theorem (Elkies) [Java]
805. Six Circles Theorem (Evelyn) [Java]
806. Six Concyclic Points [Java]
807. Six Concyclic Points II [Java]
808. Six Incircles in an Equilateral Triangle [Java]
809. Six Point Circle [Java]
810. SixPoints, ThreeLines [Java]
811. Soddy Circles and David Eppstein's Centers [Java]
812. Some Properties of Napoleon's Configuration [Java]
813. Sperner's Lemma [Java]
814. Spiral Similarity [Java]
815. Square and Circle in a Gothic Cupola
816. Square From Nowhere [Java]
817. Square in a Circle Inscribed in a Square [Java]
818. Square in a Right Triangle [Java]
819. Square in Chair [Java]
820. Square Inscribed in a Triangle [Java]
821. Square Inscribed in a Triangle II [Java]
822. Square, Similarity and Slopes [Java]
823. Squares and Straight Tetrominoes
824. Squares in Circles [Java]
825. Squares in Semicircle and Circle
826. Squares in Triangles
827. Squares Inscribed In a Triangle III [Java]
828. Squares on Sides of a Quadrilateral [Java]
     • Right Isosceles Triangles on Sides of a Quadrilateral [Java]
829. Square out of a Quadrilateral
830. Squaring a circle
     • Ancient Problem = Ancient Solution
831. Squaring a Rectangle [Java]
832. Squinting Eyes Theorem [Java]
833. SSA
834. SSS
835. Staircase Illusion [Java]
836. Star of David [Java]
837. Steiner's Sangaku
838. Steiner's Theorem
839. Stereographic Projection [Java]
840. Stereographic Projection and Inversion
841. Stereographic Projection and Radical Axes
     • Stereographic Projection of a Coffin Problem
842. Stewart's theorem
843. Straight Edge Only Construction of Polar
844. Structural Constellation [Java]
845. Sum of Squares in Equilateral Triangle
846. Sum of Squares of Distances to Vertices
847. Symmedians
     • Symmedian and Antiparallel [Java]
     • Symmedian and 2 Antiparallels [Java]
     • Symmedian and the Tangents [Java]
     • Symmedian in a Right Triangle [Java]
848. Symmetries in a Triangle
849. Tangency Point of Two Circles [Java]
850. Tangent and Secant [Java]
851. Tangent as a Radical Axis [Java]
852. Tangent Circles and an Isosceles Triangle [Java]
     • Tangent Circles and an Isosceles Triangle, a Solution [Java]
     • Tangent Circles and an Isosceles Triangle, a Second Solution [Java]
853. Tangent Circles in a Parallelogram [Java]
854. Tangent in Concurrency [Java]
855. Tangent Lines and Circles in Convex Quadrilateral [Java]
856. Tangent to two circles
857. Tangents and Diagonals in Cyclic Quadrilateral [Java]
858. Tangents, Perpendiculars and Geometric Mean [Java]
859. Tarski-Banach Decompositions
860. Taylor circle [Java]
861. Terquem's Theorem [Java]
862. Tesseract [Java]
863. The 80-80-20 Triangle
     • 1 Solution to a modified problem with angles of 40° and 50°.
     • 2 Solutions to a modified problem in the 80-80-20 triangle
     • 6 Solutions to a derivative problem in the 80-80-20 triangle
     • 10 Solutions to the classical problem in the 80-80-20 triangle
     • The 80-80-20 Triangle Problem, A Variant
864. The Chaos Game: Address Space vs IFS [Java]
865. The only rational triangle
866. Thébault's Problem I [Java]
867. Thébault's Problem II [Java]
868. Thébault's Problem III [Java]
869. Thébault's Problem III, Proof [Java]
870. Thébault's Problem IV [Java]
871. Theorem of Complete Quadrilateral [Java]
872. Theorems of Ceva and Menelaus, an Illustrated Generalization [Java]
873. Three Circles and Common Tangents
     • Three circles problem - #2
     • Three circles problem - #3
874. Three Circles through the Incenter
875. Three Concurrent Circles [Java]
876. Three Congruent Circles by Reflection [Java]
877. Three Congruent Circles by Reflection II [Java]
878. Three Congruent Circles by Reflection III [Java]
879. Three Equal Circles in a Semicircle
880. Three Incircles In a Right Triangle [Java]
881. Three Incircles in a Triangle [Java]
882. Three Isosceles Triangles [Java]
883. Three Orthogonal Circles Through Three Given Points [Java]
884. Three Parallels in a Triangle [Java]
885. Three Points on a Parabola [Java]
886. Three Pyramids are Better Than Two [Java]
887. Three regions sharing a common boundary
888. Three Similar Polygons [Java]
889. Three Similar Triangles [Java]
890. Three Similar Triangles II [Java]
891. Three Squares and Two Ellipses [Java]
892. Three Tangent Circles [Java]
893. Three Tangents, Three Chords in Ellipse [Java]
894. Three Tangent Circles Sangaku
895. Three Tangents Theorem [Java]
896. Three Tangents, Three Secants [Java]
897. Three Touching Circles [Java]
898. Tiling a Chessboard with Dominoes
899. Toothpick Construction of a Square [Java]
900. Topological preliminaries
901. Torque as a Mathematical Object
902. Touching Circles [Java]
903. Touching Circles and Concurrency [Java]
904. Touching Circles with Given Centers II [Java]
905. Train wheel quandary
906. Transitivity in Action [Java]
907. Translated Triangles [Java]
908. Translation Transform [Java]
909. Triangle ABC is right iff sin²A + sin²B + sin²C = 2
910. Triangle Areas in a Parallelogram [Java]
911. Triangle Areas in a Parallelogram II [Java]
912. Triangle Classification [Java]
913. Triangle constructions and facts [Java]
914. Triangle Sliding on Two Straight Lines [Java]
915. Triangles, Squares and Areas from Temple Geometry [Java]
916. Triangles with Equal Area [Java]
917. Triangular Billiard [Java]
918. Triangulating Squares
919. Trigonometric Form of Ceva's Theorem [Java]
920. Trigonometric Functions [Java]
921. Trigonometric Proof of the Pythagorean Theorem
922. Trigonometry by Paper Folding
923. Trisectable angle that is not constructible
924. Tucker Circles [Java]
925. Tucker Circles Through Homothety [Java]
926. Twelve Matchsticks Area Puzzle
927. Twisted Cord Illusion [Java]
928. Two Altitudes, One Midpoint
929. Two Arbelos, Two Chains
930. Two Circles and a Limit - Analytic Proof
931. Two Circles and One More [Java]
932. Two Circles and Two More [Java]
933. Two Circles in a Parallelogram [Java]
934. Two Circles Inscribed in a Parallelogram [Java]
935. Two Circles in a Square [Java]
936. Two Circles in a Square II [Java]
937. Two Circles in an Angle [Java]
938. Two Circles on Angle Bisector [Java]
939. Two Circles on a Side of a Triangle [Java]
940. Two Circumcircles And Two Pairs of Parallels [Java]
941. Two Circumcircles in Triangle [Java]
942. Two coins puzzle
943. Two Congruent Circles by Reflection [Java]
944. Two Color Coloring of the Plane [Java]
945. Two Equilateral Triangles [Java]
946. Two Homotheties in a Parallelogram [Java]
947. Two Intersecting Circles [Java]
948. Two Lines - Two Circles [Java]
949. Two Parallels in a Triangle and One More [Java]
950. Two Perpendiculars From a Point to a Line [Java]
951. Two Quadrilaterals [Java]
952. Two Quadruplets of Tangent Circles [Java]
953. Two Sangaku with Equal Incircles
954. Two Simsons in a Triangle [Java]
955. Two Squares and Another Square [Java]
956. Two Symmetric Triangles Are Directly Decomposable [Java]
     • Two Symmetric Triangles Are Directly Decomposable II [Java]
     • Two Symmetric Triangles Are Directly Decomposable III [Java]
     • Two Symmetric Triangles Are Directly Decomposable IV [Java]
957. Two Touching Circles [Java]
958. Two Triangles With Common Base and Altitude [Java]
959. Two Triples of Concurrent Circles [Java]
960. Two Triples of Similar Triangles [Java]
961. Unfolding Spirals Illusion [Java]
962. Urquhart's Theorem
     • An Elementary Proof of Urquhart's Theorem
     • Urquhart's Theorem For And By Conics
     • Urquhart's theorem - From Leningrad Mathematical Olympiad
963. USA 1975 Problem 4
964. Use of Homogeneous Coordinates for Geometrical Calculations
965. Van Obel Theorem, Determinants, and Barycentric Coordinates
966. Van Schooten's and Pompeiu's Theorems [Java]
967. Van Schooten's Locus Problem [Java]
968. Variations on the Theme of Tremas [Java]
969. Variations on the Theme of Tremas II [Java]
970. Varignon Parallelogram [Java]
971. Various Geometries
972. Vecten's Collinearity [Java]
973. Vecten's Mesh [Java]
974. Vector Addition and Subtraction [Flash]
975. Vector product
976. Versatile Theorem [Java]
977. Viewing a Statue
978. Viviani's Theorem [Java]
979. Viviani's Theorem II [Java]
980. Viviani in Isosceles Triangle [Java]
981. Viviani by Inversion
982. Volumes of Two Pyramids
983. Wallace-Bolyai-Gerwien Theorem
984. Wallace's Theorem [Java]
985. Weird curves bound normal areas
986. Weitzenböck Inequality [Java]
987. What does one see inside a spherical mirror? [Java]
988. What Is Analytic Geometry?
989. What Is Angle?
990. What Is Area?
991. What Is Ellipse?
992. What Is Line?
993. What Is Point?
994. When a Triangle is Equilateral [Java]
995. When index equals content [Java]
996. Why a Diameter Is the Longest Chord?
997. William Wallace's Proof of the Butterfly Theorem
998. Wittenbauer's Parallelogram [Java]
999. Wundt Illusion [Java]
1000. Wundt Block Illusion [Java]
1001. Y. Sawayama's Lemma [Java]
      • Jack D'Aurizio Proof of Sawayama's Lemma
1002. Y. Sawayama's Theorem
1003. Zaslavsky's theorem [Java]
1004. Zöllner's Illusion [Java]

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