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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 1995 Problem 3
74. APMO 1998, Problem 4
75. APMO 2002, Problem 3
76. APMO 2003, Problem 2
77. APMO 2004, Problem 2
78. Apollonian Circles Theorem [Java]
79. Apollonian Gasket [Java]
80. Apollonius Problem [Java]
81. Approximate Construction of Regular Pentagon by A. Dürer
82. 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]
83. 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
84. Archimedes' Law of the Lever [Java]
85. Archimedes' Method
86. Archimedes' Twins in the Edo Period
87. Area of Median Triangle [Java]
88. Area of Parallelogram [Java]
89. Area of the Union of Two Squares [Java]
90. Area of Triangle [Java]
91. Areas and Centroid in a Triangle [Java]
92. Arithmetic-Geometric Mean Inequality
93. Arithmetic Mean Sangaku [Java]
94. Arithmetic Mean Sangaku II [Java]
95. Around the Incircle [Java]
96. Around the Incircle in a Polygon [Java]
97. Artisans vs Mathematicians
98. Assimilation Illusion [Java]
99. Asymmetric Propeller [Java]
100. Barbier, The Theorem of [Java]
101. Barycentric coordinates
     • Barycentric coordinates and Geometric Probability
     • Barycentric Coordinates: a Tool [Java]
102. Bender: A Visual Illusion [Java]
103. Between Major and Minor Circles [Java]
104. Bevan's Point and Theorem [Java]
105. Bimedians in a Regular Tetrahedron [Java]
106. Bisectal Circle [Java]
107. Bisecting arcs
108. Bisecting a shape
109. Bisector of an imaginary angle may be real
110. Bottema Shatters Japan's Seclusion [Java]
111. Bottema's theorem [Java]
     • On Bottema's Shoulders [Java]
     • On Bottema's Shoulders II [Java]
112. 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]
113. Bounded Distance
114. Brahmagupta's Theorem [Java]
115. Brianchon in Ellipse [Java]
116. Brianchon's theorem [Java]
117. Brick Wall Illusion [Java]
118. Bride's Chair [Java]
119. Brief introduction to Rational Trigonometry
120. 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]
121. Building a Bridge [Java]
122. Building Bridges [Java]
123. Bulging lines illusion [Java]
124. 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]
125. Cabart's Collinearity
126. Cantor Set and Function
     • Cantor sets [Java]
     • C₀ - C₀ = [-1, 1]
127. Cantor's Theorem [Java]
128. Carnot's Theorem
129. Carnot's Theorem
130. Carnot's Theorem (Generalization of Wallace's theorem) [Java]
131. Carpet With a Hole [Java]
132. 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]
133. Centroid, a Characteristic Property Of
134. Centroids in a Polygon [Java]
135. Chasing Angles in Pascal's Hexagon [Java]
136. Circles Tangent to Circumcircle [Java]
137. Circles with Tangent Diameters [Java]
138. Ceva and Menelaus Meet on the Roads
139. Ceva in Circumscribed Quadrilateral
140. Ceva's Theorem
141. Ceva's Theorem and Fibonacci Bamboozlement [Java]
     • A Faulty Dissection [Java]
142. Cevian Nest [Java]
143. Cevian Triangle [Java]
144. Chain of Inscribed Circles [Java]
145. Chaos, Emergence of [Java]
146. Chasles' Theorem
     • Chasles' Theorem, a Proof [Java]
     • Chasles' Theorem, a Simple Proof
147. Cherchez le quadrilatère cyclique [Java]
148. Cherchez le quadrilatère cyclique II [Java]
149. Chords, Concurrency and Orthic Triangle [Java]
150. Chvatal's Art Gallery Theorem [Java]
151. Circle and Hyperbola as Lighthouse Curves
152. Circle and Hyperbola as Lighthouse Curves II
153. Circle Crosses Triangle [Java]
154. Circle Concurrency and Spiral Similarity [Java]
155. Circle Division by Euclid and Leonardo [Java]
156. Circle of Similitude [Java]
157. Circle-Stacking Theorem [Java]
158. Circles And Parallels [Java]
159. Circles and Semicircles in Rectangle [Java]
160. Circles in a Circular Segment
161. Circles in Morley's Triangles [Java]
162. Circles Lined on the Legs of a Right Triangle [Java]
163. Circles On Cevians [Java]
164. Circles through the Orthocenter [Java]
165. Circles with Equal Collinear Chords [Java]
166. Circular Poggendorff Illusion [Java]
167. Circle Inscribed in a Circular Segment [Java]
168. Circumcenter on Angle Bisector [Java]
169. Circumcevian Triangle [Java]
170. Classification of Quadrilaterals [Java]
171. Cleavance Center [Java]
172. Cleaver [Java]
173. Clifford's Chain [Java]
174. Coaxal Circles Theorem [Java]
175. Collage Theorem [Java]
176. Collinearity in Bicentric Quadrilaterals [Java]
177. Color Cycling on the Mandelbrot Set [Java]
178. Common Chord and a Tangent [Java]
179. Common Tangents to Two Circles I [Java]
180. Common Tangents to Two Circles II [Java]
181. The Complete Quadrilateral [Java]
182. Complex Numbers and Geometry
183. Concurrency in the Intouch Triangle [Java]
184. Concyclic Circumcenters: A Dynamic View [Java]
     • Concyclic Circumcenters: A Sequel [Java]
185. Concyclic Points in a Triangle
186. Concyclic Points in Equilateral Bumps [Java]
187. Conic Sections
188. Conic Sections as Loci of Points [Java]
189. Conjugate Diameters in Ellipse [Java]
190. Consecutive Isosceles Decomposition [Java]
191. Constant Chord
192. Constructing a triangle from its angle bisectors is in general impossible
193. Construction of a Triangle from the Circumcenter, Orthocenter and Incenter
194. Construction of a Triangle from Two Vertices and the Centroid
195. Construction of Conics from Pascal's Theorem [Java]
196. Construction of Regular Pentagon by H. W. Richmond
197. Construction of the Angle Bisector [Java]
198. Construction of the Perpendicular Bisector [Java]
199. Construction problem [Java]
200. Construction with the Compass Only
201. Continuous Line Illusion [Java]
202. Convex Polygon Is the Intersection of Half Planes [Java]
203. Copernicus' Theorem [Java]
     • Proof of Copernicus' Theorem [Java]
204. cos(36°) = (1 + √5)/4
205. 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]
206. Counting Triangles and Segments in a Polygon [Java]
207. Cross-Ratio
208. Crossed-Lines Construction of Shapes of Constant Width [Java]
209. Crossing Number of a Graph
210. Cube Counting Ambiguity [Java]
211. Pencils of Cubics
212. Curry's Paradox [Java]
213. Cut the Cone [Java]
214. Cut the Cube [Java]
215. Cut the Cylinder [Java]
216. Cutting Squares
217. Cyclic Hexagon [Java]
218. Cycloids [Java]
219. Day's Sine Illusion [Java]
220. Delboeuf Illusion [Java]
221. Delian Problem Solved [Java]
222. Desargues in the Bride's Chair (with Pythagoras)
223. Desargues' Theorem [Java]
224. Determination of p
225. Diagonal Count [Java]
226. Diagonals in a Cyclic Quadrilateral
227. Diameters and Chords [Java]
228. Directly Similar Figures [Java]
229. Disappearing Lines puzzle [Java]
230. Discovery of Duane DeTemple [Java]
231. Dissection of a 10×13 Rectangle into Two Chessboards
232. Dissection of Cyclic Quadrilateral [Java]
233. Distance
234. Distance between the Orthocenter and Circumcenter
235. Distance Formula
236. Distorted Lines Illusion [Java]
237. Divergent Spirals Illusion [Java]
238. Divide a Circle into N Parts of Equal Area
239. 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]
240. Dividing Evenly a Quadrilateral [Java]
241. Dividing Evenly a Quadrilateral II [Java]
242. Double Generation Theorem [Java]
243. Drawing circles [Java]
244. Droz-Farny Circles [Java]
245. Droz-Farny Line Theorem [Java]
246. Duplication of the Cube
247. Dynamic Construction of Ellipse and Other Curves [Java]
248. E. W. Dijkstra's Proof of the Pythagorean Theorem [Java]
249. Easy Construction of Bicentric Quadrilateral [Java]
250. Easy Construction of Bicentric Quadrilateral II [Java]
251. Egyptian Triangle By Paper Folding
252. Eight Equal Tangents [Java]
253. Eight Point Circle [Java]
254. Ellipse in Arbelos [Java]
255. Ellipse Between Two Circles [Java]
256. Equal Angles in Two Circles [Java]
257. Equal Angles in Two Circles II [Java]
258. Equal Areas In Parallelogram à la Pythagoras [Java]
259. Equal Chords in Crossing Circles [Java]
260. Equal Circles, Medial Triangle and Orthocenter [Java]
261. Equal Incircles Theorem [Java]
262. Equichordal points: Just Do It [Java]
263. Equidecomposition of a Rectangle and a Square [Java]
264. Equidecomposition of Two Rectangles [Java]
265. Equidecomposition of a Triangle and a Rectangle [Java]
266. Equidecomposition of a Triangle and a Rectangle II [Java]
267. Equidecomposition of Two Parallelograms [Java]
268. Equilateral and 3-4-5 Triangles [Java]
269. Equilateral Triangle on Parallel Lines [Java]
270. Equilateral Triangle, Straight Line and Tangent Circles [Java]
271. Equilateral Triangles and Incircles in a Square
272. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral [Java]
273. Equilateral Triangles On Sides of a Parallelogram [Java]
274. Equilateral Triangles On Sides of a Parallelogram II [Java]
275. Equilateral Triangles on Sides of a Quadrilateral [Java]
276. Equilic Quadrilateral I, a Flawed Proof [Java]
277. Equilic Quadrilateral I, A Variation [Java]
278. Equilic Quadrilateral II [Java]
279. The Erdös-Mordell Inequality
280. Escher's Theorem [Java]
281. Euclid's Fifth Postulate
     • Similarity and the Parallel Postulate
282. Euclid's Elements Reference Page
283. Euclidean Construction of Center of Ellipse [Java]
284. Euclidean Construction of Tangent to Ellipse
285. Euler Line Cuts Off Equilateral Triangle [Java]
286. Euler Line and 9-Point Circle [Java]
287. Euler's Formula and Poncelet Porism
288. Existence of the Circumcenter [Java]
289. Experimentation with Dynamic Geometry Software: An Example [Java]
290. Euclid I.43 [Java]
291. Euclid I.43 Extended [Java]
292. Exterior Angle Theorem - an appreciation
293. External Angle Bisector [Java]
294. Extouch Triangle in Poncelet Porism [Java]
295. Eyeball Theorem [Java]
296. Fagnano's problem [Java]
     • Fagnano's Problem, Third Solution [Java]
     • Fagnano's Problem, Izvolsky's Solution [Java]
     • Fagnano's Problem In Reverse [Java]
297. 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]
298. Feuerbach's Theorem: A Proof [Java]
299. Find a Common Chord of Given Length [Java]
300. Find the Angle! [Java]
301. Finding Circle Center By Ruler Alone [Java]
302. Five Concyclic Points [Java]
303. Five Incircles in a Square [Java]
304. Five Incircles Theorem [Java]
305. Fixed Point of Circles Orthogonal to the Given One [Java]
306. Fleury's Algorithm and Euler's Paths and Cycles [Java]
307. Foot of Altitude and Minimum Distance [Java]
308. Focal Definition of Ellipse [Java]
309. Focus and Directrix of Ellipse [Java]
310. For every object there is a distance at which it looks its best.
311. Ford's touching circles
312. Four Centroids and Parallels [Java]
313. Four Circles In a Triangle [Java]
314. Four Concurrent Lines in a Cyclic Quadrilateral [Java]
315. Four Construction Problems [Java]
316. Four Geometric Problems of Antiquity
317. Four Hinged Squares [Java]
318. Four Incident Circles [Java]
319. Four Incircles in Equilateral Triangle [Java]
320. Four 9-Point Circles in a Quadrilateral [Java]
321. Four Pedal Circles [Java]
322. Four Touching Circles [Java]
323. Four Touching Circles II [Java]
324. Four Touching Circles III [Java]
325. Four Triangles, One Circle [Java]
326. Four Turtles (or Bugs, or Dogs, etc.) [Java]
327. Fractal curves and dimension [Java]
328. Friendly Kiepert's Perspectors [Java]
329. From Foci to a Tangent in Ellipse [Java]
330. Fundamental Theorem of 3-Bar Motion [Java]
331. Fuss' Theorem [Java]
332. Generalizing a Romanian Olympiad Problem [Java]
333. Geometric Illustration of a Convergent Series
334. Geometric Mean Sangaku
335. Geometric Meaning of the Geometric Mean
336. Geometric Optimization from the Asian Pacific Mathematical Olympiad [Java]
337. Gergonne and Medial Triangles Are Orthologic [Java]
338. Gergonne and Soddy Lines Are Perpendicular [Java]
339. Gergonne in Ellipse [Java]
340. Gergonne and Van Obel Theorems
341. Gion Shrine Problem
342. Given Parabola, Find Axis
343. Glide Reflection [Java]
344. Golden Ratio in Geometry
345. Golden Ratio in an Irregular Pentagon
     • Golden Ratio in an Irregular Pentagon, Construction I [Java]
     • Golden Ratio in an Irregular Pentagon, Construction II [Java]
346. Golomb's inductive proof of a tromino theorem [Java]
     • Tromino as a Rep-tile [Java]
347. Gothic Arc [Java]
348. Graph Theory [Java]
349. Grebe: from Ladies' Diary to Carroll's Pillow [Java]
350. Grebe's Theorem
351. Greg Markowsky's Problem for Parabola [Java]
352. Griffiths Points in Cyclic Quadrilateral [Java]
353. Half Turn, Reflection in Point [Java]
354. Halving a square
355. Harmonic Mean Sangaku [Java]
356. Harmonic Ratio
357. Harmonic Ratio in Complex Domain
358. Hart Circle [Java]
359. Hart's Inversor [Java]
360. Haruki's Lemma [Java]
361. Haruki's Theorem
362. Hausdorff Distance
363. Hearty Munching on Cardioids [Java]
364. Height/Width Interplay Illusion [Java]
365. Hering Illusion [Java]
366. Hermann Grid Illusion [Java]
367. Heron's formula
368. Heron's Problem [Java]
369. Hexagon Parallel to Medial Triangle [Java]
370. Hexagon Parallel to Orthic Triangle [Java]
371. Hinged Squares [Java]
372. Hippocrates' Squaring of a Lune [Java]
373. Hippocrates' Squaring of Lunes
374. Hjelmslev Theorem [Java]
375. Homologous Lines under Three Spiral Similarities [Java]
376. Homothety [Java]
377. Hooper's Paradox [Java]
378. Hunting Right Angles [Java]
379. Ian McGee's Observation [Java]
380. IMO 2004, Problem 5
381. IMO 2006, Problem 1
382. Impossible Fork [Java]
383. Impossible Frame [Java]
384. Importance of Occluders: Motion Binding [Java]
385. In the Wasan Spirit
386. Incenters in Cyclic Quadrilateral [Java]
387. Incidence in Feuerbach's Theorem [Java]
388. Infinitude of Primes - A Topological Proof
389. Inscribed and Central Angles in a Circle [Java]
390. Inscribed Angles [Java]
391. Inscriptible and Exscriptible Quadrilaterals [Java]
392. Inscriptible Quadrilateral: An Illustration [Java]
393. Internal Tangents to Three Circles [Java]
394. Intersecting Chords Theorem
395. Intouch Triangle in Poncelet Porism [Java]
396. Invariant Area Sweep Implies Pythagorean Theorem [Java]
397. Inversion: Reflection in a Circle [Java]
     • Inversion [Java]
     • Inversion in the Incircle [Java]
     • Inversion Tool
     • Inversion with Negative Power [Java]
398. Is Every Trapezoid Parallelogram? [Java]
399. Isogonal and Isotomic Conjugalities [Java]
400. Isogonal Concurrency [Java]
401. Isogonal image of the circumcircle [Java]
402. Isoperimetric Theorem and Inequality
     • Isoperimetric Property of Equilateral Triangles
     • Isoperimetric Theorem for Rectangles
     • A Variant of the Isoperimetric Theorem
403. Isosceles on the Sides of a Triangle [Java]
404. Isotomy and Isogonality Hand-in-Hand [Java]
405. Iterations and the Mandelbrot Set [Java]
406. Iterations on a Circle Through Three Points [Java]
407. Iterations on Euler Lines [Java]
408. Jan Stevens' proof of E. W. Dijkstra's Generalization of the Pythagorean theorem
409. Joachimsthal's Notations
410. John Sharp's Paradox [Java]
411. Join Circles by Given Segment [Java]
412. Judd Illusion [Java]
413. Julia sets indexing [Java]
414. Kanizsa Triangle [Java]
415. Kiepert's Theorem [Java]
416. Knots [avi movie]
417. Kürschak's Tile and Theorem [Java]
418. La Hire's Theorem [Java]
     • La Hire's Theorem [Java]
     • La Hire's Theorem in Ellipse [Java]
419. Langman's Paradox [Java]
420. Lean Napoleon's Triangles [Java]
421. Length of the diagonal of the unit square equals the square root of 2
422. Lepidoptera of the Circles [Java]
423. Line IO in Bicentric Quadrilaterals
424. Line through a Point in an Angle
425. Locus of Points in a Given Ratio to Two Points [Java]
426. Longest segment
427. MacLaurin's Construction of Conics [Java]
428. Maimed Cake [Java]
429. Malfatti's Problem [Java]
430. Malfatti's Problem, Hart's Solution [Java]
431. Mathematical Droodles [Java]
432. Mathematics in Pizzeria [Java]
433. MathPro Logo [Java]
434. Maximal Properties of the Pythagorean Relation
435. Maximum Area Property of Equilateral Triangles
436. Maxwell's Theorem [Java]
     • Maxwell Theorem via the Center of Gravity
     • Projective Proof of Maxwell's Theorem
     • Projective Generalization of Maxwell's Theorem [Java]
437. May the Product of Planes Be a Sphere?
438. Mechanical Proofs of the Pythagorean Theorem
439. Median in Touching Circles [Java]
440. Medians
441. Medians in a Quadrilateral [Java]
442. Menelaus Theorem [Java]
     • Menelaus Theorem: Proofs Ugly and Elegant - A. Einstein's View [Java]
     • Menelaus From Ceva [Java]
443. Metric Relations in a Triangle
444. Midline in Similar Triangles [Java]
445. Midpoints from Gergonne Triangle [Java]
446. Midpoints of the Lines Joining In- and Excenters [Java]
447. Minesweeper Theorem
448. Miquel's Point [Java]
449. Miquel's Point of a 4-line Via Spiral Similarity [Java]
450. Miquel's Theorem for Circles [Java]
451. Mirror on the Wall [Java]
452. Mirror property of the altitudes [Java]
453. Mirror Property of Altitudes via Pascal's Hexagram [Java]
454. Mixtilinear Circles and Concurrence [Java]
455. Moebius Strip [avi movie]
456. Möbius' Strip [Java]
457. Models and Metamathematics
458. Monge via Desargues [Java]
459. Monge via Desargues II [Java]
460. Morley Constellation [Java]
461. 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
462. Morphing [Java]
463. Müller-Lyer Illusion [Java]
464. Munching on Circles
465. Munching on Inscribed Angles
     • More On Inscribed Angles [Java]
466. My logo
467. Nagel Line [Java]
468. Nagel point [Java]
469. Nagel Point of the Medial Triangle [Java]
470. Nagel's Theorem [Java]
471. 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]
472. Necker Cube [Java]
473. Neuberg Sangaku [Java]
474. Newton's Construction of Conics [Java]
475. Newton's Theorem [Java]
476. Nine Point Circle [Java]
477. No Equilateral Triangles, Please [Java]
478. Nobbs' Points, Gergonne Line [Java]
479. Non-Eclidean Geometries
480. No-Pedal Collinearity [Java]
481. Occluded Wheel Illusion [Java]
482. Olof Hanner's Jigsaw Puzzle [Java]
483. On the Difference of Areas
484. Optimization Problem in Acute Angle [Java]
485. Octagon in Parallelogram: Overlapping Regions [Java]
486. Octagon in Parallelogram [Java]
487. Optical Property of Ellipse [Java]
488. Orbison illusion [Java]
489. Orthocenter and Three Equal Circles [Java]
490. Orthocenters
491. Orthocenters of Two Triangles Sharing Circumcenter and Base [Java]
492. Orthocentric System From Rectangle [Java]
493. Orthodiagonal and Cyclic Quadrilaterals [Java]
494. Orthogonality in Isosceles Triangles [Java]
495. Orthologic Triangles in a Quadrilateral [Java]
496. Orthopole [Java]
497. Out of Pentagon Sangaku
498. Packing shapes in a plane
499. Pairs of Homologous Lines under Spiral Similarities [Java]
500. Pairs of Incircles in a Quadrilateral [Java]
501. Paper Folding & Cutting Sangaku
502. Paper Folding Geometry
503. Paper strip activities [avi movies]
504. Paper strip activities, two strips
505. Pappus' Generalization of Euclid I.47 [Java]
506. Pappus' Theorem [Java]
     • Dual to Pappus'Theorem [Java]
507. Parabola [Java]
508. Paragon [Java]
509. Parahexagon [Java]
510. Parallel Chords [Java]
511. Parallel Chords II [Java]
512. Parallel Chords in Ellipse [Java]
513. Parallel Chords in Crossing Circles [Java]
514. Parallel Lines in a Quadrilateral [Java]
515. Parallelogram and Similar Triangles [Java]
516. Parallelogram Illusion [Java]
517. Parallelogram Law [Java]
518. Parallelogram Law: A PWW [Java]
519. Parallelogram Law: A Tessellation [Java]
520. Partition of point sets in a plane
521. Partitioning 3-Space with Circles
522. Partitioning a Circle
523. Pascal: Necessary and Sufficient [Java]
524. Pascal in a Cyclic Quadrilateral [Java]
525. Pascal Lines: Steiner and Kirkman Theorems [Java]
526. Pascal Lines: Steiner and Kirkman Theorems II [Java]
527. Pascal's theorem [Java]
     • Pascal's Theorem, Homogeneous Coordinates
     • Illustration of Pascal's Theorem [Java]
     • Projective Proof of Pascal's Theorem [Java]
528. Peacock's Tail Sangaku
529. Pedal Collinearities [Java]
530. Pedal Polygons [Java]
531. Pedal Triangle and Isogonal Conjugacy [Java]
532. Pedoe's Theorem [Java]
533. Pentagon, construction of
     • Regular Pentagon Construction by Y. Hirano
534. Pentagon Proportions Sangaku
535. Periodic patterns in sin(N) [Java]
536. Perpendicular Bisectors in an Inscriptible Quadrilateral [Java]
537. Perpendicular Bisectors in an Inscriptible Quadrilateral II [Java]
538. Pivot Theorem [Java]
539. Plane Filling Curves [Java]
     All Peano Curves [Java]
540. Plane Isometries
541. Plato's Geometric Numbers: can you better Sam Loyd?
542. Peaucellier Linkage [Java]
543. Pick's Theorem [Java]
544. Poggendorff's Illusion
545. Point common to two similar rectangles [Java]
546. Point in a Square
547. Polar Circle [Java]
548. Poles and Polars [Java]
     • Poles, Polars and Quadrilaterals [Java]
     • Pole and Polar with respect to a Triangle [Java]
549. Polygons: formality and intuition [Java]
550. Polygons Formed by Perpendicular Bisectors [Java]
551. Poncelet's Porism [Java]
552. Poncelet's Porism in Ellipses [Java]
553. Poncelet's Theorem [Java]
554. Ponzo Illusion [Java]
555. Ponzo Illusion II [Java]
556. Possible or Impossible? [Java]
557. Power of a point with respect to a circle
558. Problem 4, 1975 USA Math Olympiad and the Radical Axis [Java]
559. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles [Java]
560. Problem 4, 1975 USA Math Olympiad: Normals and Tangents [Java]
561. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch [Java]
562. Problem 4 from the IMO 2009
563. Problem in an Equilateral Triangle [Java]
564. Problem in an Equilateral Triangle II [Java]
565. Problem in Three Squares [Java]
566. Problem of Equal Steps [Java]
567. Problem of Equal Steps II [Java]
568. Problem on a Pentagonal Star
569. Projections on Internal and External Angle Bisectors [Java]
570. Projective Collinearity in a Quadrilateral [Java]
571. Projective infinity
572. Properties of the Circle of Similitude [Java]
573. Property of 6-Parpolygon [Java]
574. Property of Angle Bisectors [Java]
575. Property of Angle Bisectors II [Java]
576. Property of Circumscribed Quadrilaterals [Java]
577. Property of Points Where In- and Excircles Touch a Triangle [Java]
578. Ptolemy's Theorem
579. Ptolemy on Hinges
580. Puzzles on graphs.
581. 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?
582. Pythagoras and Vecten Break Japan's Isolation [Java]
583. Pythagorean Triples [Java]
584. Quadrature: A Child's Play
585. Quadrilateral With Equal Opposite Sides And Angles [Java]
586. Quadrilaterals Formed by Perpendicular Bisectors [Java]
587. Quest for Paragon [Java]
588. Radical Axis and Center, an Application [Java]
589. Radical Axis of Circles Inscribed in a Circular Segment [Java]
590. Radical Center [Java]
591. Radius and Construction of a Mixtilinear Circle [Java]
592. Radius of a Circle by Paper Folding [Java]
593. Random Dot Stereograms [Java]
594. Rational Points On a Circle
595. Reflection In Line [Java]
596. Reflections in Ellipse [Java]
597. Reflections of a Line Through the Orthocenter [Java]
598. Reflections of a Point on the Circumcircle [Java]
599. Reflections of the Orthocenter [Java]
600. Regular Polyhedra - Platonic Solids
     • Tetrahedron [Java]
     • Octahedron [Java]
     • Cube [Java]
     • Icosahedron [Java]
     • Dodecahedron [Java]
601. Regular polyhedra and Euler's Theorem
602. Relations in a Cyclic Polygon
603. Remarkable Line in a Quadrilateral [Java]
604. Remarkable Line in Cyclic Quadrilateral [Java]
605. Rendezvous [Java]
606. Reverse Spoke Illusion [Java]
607. Revolving Circles Illusion [Java]
608. Rhombus in a Cyclic Quadrilateral [Java]
609. Riemann Sphere and Möbius Transformation
610. Rotating Top Illusion [Java]
611. Rotating Gears [Java]
612. Rotation Transform [Java]
     • Product of Rotations [Java]
     • Rotations in Disguise [Java]
613. Rouse Ball's Fallacy [Java]
614. Rusty Compass Construction of Equilateral Triangle [Java]
615. Salinon: From Archimedes' Book of Lemmas [Java]
616. Sam Loyd's Geometric Puzzle
617. Sam Loyd's Son's Dissection [Java]
618. Sangaku: Reflections and the Phenomenon
619. Sangaku: Critique of My View and a Response
620. Sangaku and The Egyptian Triangle
621. Sangaku in a Square
622. Sangaku Iterations, Is it Wasan? [Java]
623. Sangaku with 8 Circles
624. Sangaku with Quadratic Optimization
625. Sangaku with Three Mixtilinear Circles [Java]
626. Sangaku with Versines [Java]
627. Sangakus with a Mixtilinear Circle [Java]
628. Scintillating Grid Illusion [Java]
629. Secant Angles [Java]
630. Secant Angles II [Java]
631. Secant, Normal, Tangent [Java]
632. Secant, Tangents and Orthogonality [Java]
633. See-Saw Lemma [Java]
634. Segment Trisection [Java]
635. Separating point sets with a circle
636. Sequences of Touching Circles
637. Seven Circles Theorem [Java]
638. Shaggy Dog Theorem
639. Shapes in a lattice
640. Shapes of constant width [Java]
     • Star Construction of Shapes of Constant Width [Java]
641. Shearing and Translation in Pythagorean Pants [Java]
642. Shearing a Polygon into a Triangle of Equal Area [Java]
643. Shearing Butterflies in Quadrilaterals [Java]
644. Shepard's Parallelogram Illusion [Java]
645. Shifting Cube Illusion [Java]
646. Shimmer Illusion [Java]
647. Shortest Fence in a Quarter-Circle Pasture
648. Shredding the torus
649. Sierpinski Gasket By Common Trema Removal [Java]
650. Sierpinski Gasket Via Chaos Game [Java]
651. Sierpinski Gasket by Trema Removal [Java]
652. Similar Triangles And More
653. Similar Triangles on Sides of a Quadrilateral [Java]
654. Similar Triangles on Sides and Diagonals of a Quadrilateral [Java]
655. Simple Quadrilaterals Tessellate the Plane [Java]
656. Simson Line [Java]
     • Simson Line From Isogonal Perspective [Java]
     • Simson Lines Of Diametrically Opposite Points [Java]
657. Simsons and 9-Point Circles in Cyclic Quadrilateral [Java]
658. Simultaneous Contrast Illusion [Java]
659. Simultaneous Contrast Illusion II [Java]
660. Simultaneous Contrast: Koffka's Illusion [Java]
661. Simultaneous Contrast: White's Illusion [Java]
662. Simultaneous Contrast: White's Illusion, V-Variant [Java]
663. Simultaneous Generalization of the Theorems of Ceva and Menelaus
664. Sine, Cosine, and Ptolemy's Theorem
665. Six Circles Theorem (Elkies) [Java]
666. Six Circles Theorem (Evelyn) [Java]
667. Six Concyclic Points [Java]
668. Six Concyclic Points II [Java]
669. Six Incircles in an Equilateral Triangle [Java]
670. Six Point Circle [Java]
671. SixPoints, ThreeLines [Java]
672. Soddy Circles and David Eppstein's Centers [Java]
673. Some Properties of Napoleon's Configuration [Java]
674. Sperner's Lemma [Java]
675. Spiral Similarity [Java]
676. Square and Circle in a Gothic Cupola
677. Square From Nowhere [Java]
678. Square in a Right Triangle [Java]
679. Square in Chair [Java]
680. Square Inscribed in a Triangle [Java]
681. Square Inscribed in a Triangle II [Java]
682. Squares in Circles [Java]
683. Squares Inscribed In a Triangle III [Java]
684. Squares on Sides of a Quadrilateral [Java]
     • Right Isosceles Triangles on Sides of a Quadrilateral [Java]
685. Squaring a circle
686. Squaring a Rectangle [Java]
687. Squinting Eyes Theorem [Java]
688. SSA
689. SSS
690. Staircase Illusion [Java]
691. Star of David [Java]
692. Steiner's Sangaku
693. Steiner's Theorem
694. Stereographic Projection [Java]
695. Stereographic Projection and Inversion
696. Stereographic Projection and Radical Axes
     • Stereographic Projection of a Coffin Problem
697. Stewart's theorem
698. Straight Edge Only Construction of Polar
699. Structural Constellation [Java]
700. Sum of Squares in Equilateral Triangle
701. Sum of Squares of Distances to Vertices
702. Symmedians
     • Symmedian and Antiparallel [Java]
     • Symmedian and 2 Antiparallels [Java]
     • Symmedian and the Tangents [Java]
     • Symmedian in a Right Triangle [Java]
703. Symmetries in a Triangle
704. Tangency Point of Two Circles [Java]
705. Tangent and Secant [Java]
706. Tangent as a Radical Axis [Java]
707. 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]
708. Tangent Circles in a Parallelogram [Java]
709. Tangent in Concurrency [Java]
710. Tangent to two circles
711. Tangents and Diagonals in Cyclic Quadrilateral [Java]
712. Tangents, Perpendiculars and Geometric Mean [Java]
713. Tarski-Banach Decompositions
714. Taylor circle [Java]
715. Tesseract [Java]
716. 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
717. The Chaos Game: Address Space vs IFS [Java]
718. The only rational triangle
719. Thébault's Problem I [Java]
720. Thébault's Problem II [Java]
721. Thébault's Problem III [Java]
722. Thébault's Problem III, Proof [Java]
723. Thébault's Problem IV [Java]
724. Theorem of Complete Quadrilateral [Java]
725. Theorems of Ceva and Menelaus, an Illustrated Generalization [Java]
726. Three Circles and Common Tangents
     • Three circles problem - #2
     • Three circles problem - #3
727. Three Concurrent Circles [Java]
728. Three Congruent Circles by Reflection [Java]
729. Three Congruent Circles by Reflection II [Java]
730. Three Congruent Circles by Reflection III [Java]
731. Three Incircles In a Right Triangle [Java]
732. Three Incircles in a Triangle [Java]
733. Three Isosceles Triangles [Java]
734. Three Parallels in a Triangle [Java]
735. Three Points on a Parabola [Java]
736. Three Pyramids are Better Than Two [Java]
737. Three regions sharing a common boundary
738. Three Similar Polygons [Java]
739. Three Similar Triangles [Java]
740. Three Similar Triangles II [Java]
741. Three Squares and Two Ellipses [Java]
742. Three Tangent Circles [Java]
743. Three Tangents, Three Chords in Ellipse [Java]
744. Three Tangent Circles Sangaku
745. Three Tangents Theorem [Java]
746. Three Tangents, Three Secants [Java]
747. Three Touching Circles [Java]
748. Toothpick Construction of a Square [Java]
749. Topological preliminaries
750. Touching Circles [Java]
751. Touching Circles and Concurrency [Java]
752. Train wheel quandary
753. Transitivity in Action [Java]
754. Translated Triangles [Java]
755. Translation Transform [Java]
756. Triangle Areas in a Parallelogram [Java]
757. Triangle Areas in a Parallelogram II [Java]
758. Triangle Classification [Java]
759. Triangle constructions and facts [Java]
760. Triangle Sliding on Two Straight Lines [Java]
761. Triangles, Squares and Areas from Temple Geometry [Java]
762. Triangles with Equal Area [Java]
763. Triangular Billiard [Java]
764. Triangulating Squares
765. Trigonometric Form of Ceva's Theorem [Java]
766. Trigonometric Functions [Java]
767. Trigonometric Proof of the Pythagorean Theorem
768. Trigonometry by Paper Folding
769. Trisectable angle that is not constructible
770. Tucker Circles [Java]
771. Tucker Circles Through Homothety [Java]
772. Twelve Matchsticks Area Puzzle
773. Twisted Cord Illusion [Java]
774. Two Arbelos, Two Chains
775. Two Circles and One More [Java]
776. Two Circles and Two More [Java]
777. Two Circles in a Parallelogram [Java]
778. Two Circles Inscribed in a Parallelogram [Java]
779. Two Circles in a Square [Java]
780. Two Circles in a Square II [Java]
781. Two Circles in an Angle [Java]
782. Two Circles on a Side of a Triangle [Java]
783. Two Circumcircles And Two Pairs of Parallels [Java]
784. Two Circumcircles in Triangle [Java]
785. Two coins puzzle
786. Two Congruent Circles by Reflection [Java]
787. Two Color Coloring of the Plane [Java]
788. Two Homotheties in a Parallelogram [Java]
789. Two Intersecting Circles [Java]
790. Two Parallels in a Triangle and One More [Java]
791. Two Perpendiculars From a Point to a Line [Java]
792. Two Quadrilaterals [Java]
793. Two Quadruplets of Tangent Circles [Java]
794. Two Sangaku with Equal Incircles
795. Two Simsons in a Triangle [Java]
796. Two Squares and Another Square [Java]
797. 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]
798. Two Touching Circles [Java]
799. Two Triangles With Common Base and Altitude [Java]
800. Two Triples of Similar Triangles [Java]
801. Unfolding Spirals Illusion [Java]
802. Urquhart's Theorem
     • An Elementary Proof of Urquhart's Theorem
     • Urquhart's Theorem For And By Conics
     • Urquhart's theorem - From Leningrad Mathematical Olympiad
803. USA 1975 Problem 4
804. Use of Homogeneous Coordinates for Geometrical Calculations
805. Van Obel Theorem, Determinants, and Barycentric Coordinates
806. Van Schooten's and Pompeiu's Theorems [Java]
807. Variations on the Theme of Tremas [Java]
808. Variations on the Theme of Tremas II [Java]
809. Varignon Parallelogram [Java]
810. Various Geometries
811. Vecten's Collinearity [Java]
812. Vecten's Mesh [Java]
813. Vector Addition and Subtraction [Flash]
814. Vector product
815. Versatile Theorem [Java]
816. Viewing a Statue
817. Viviani's Theorem [Java]
818. Viviani's Theorem II [Java]
819. Wallace-Bolyai-Gerwien Theorem
820. Wallace's Theorem [Java]
821. Weird curves bound normal areas
822. Weitzenböck Inequality [Java]
823. What does one see inside a spherical mirror? [Java]
824. What Is Analytic Geometry?
825. What Is Angle?
826. What Is Ellipse?
827. When a Triangle is Equilateral [Java]
828. When index equals content [Java]
829. Why a Diameter Is the Longest Chord?
830. Wittenbauer's Parallelogram [Java]
831. Wundt Illusion [Java]
832. Wundt Block Illusion [Java]
833. Y. Sawayama's Lemma [Java]
834. Y. Sawayama's Theorem
835. Zaslavsky's theorem [Java]
836. Zöllner's Illusion [Java]

Copyright © 1996-2010 Alexander Bogomolny