Geometry articles, theorems, problems.

What Is Geometry?
What Is Analytic Geometry?

Pictorial Geometry Index

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

Copyright © 1996-2010 Alexander Bogomolny