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

Copyright © 1996-2009 Alexander Bogomolny