Geometry articles, theorems, problems.

What Is Geometry?
What Is Analytic Geometry?

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

Copyright © 1996-2008 Alexander Bogomolny