Interactive Mathematics Miscellany and Puzzles

Geometry Page

What Is Geometry?
What Is Analytic Geometry?

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

Copyright © 1996-2008 Alexander Bogomolny