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

Copyright © 1996-2009 Alexander Bogomolny