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Manifesto  |  Bookstore  |  Contents  |  Amazon store  |  Term index  |  What changed?  |  Contact  |  Recommend
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Memory and MatchingArithmeticAlgebra
LogicFractalsProbability
Visual IllusionsCalculusMath Magic
CombinatoricsCombinatorial GamesPuzzles & Games
Geometric FallaciesSocial SciencesElementary Geometry
StereometryOutline MathematicsMiscellaneous Demonstrations
Interactive Mathematics Activities

    Memory and Matching

  1. Match answers
  2. Memory - Easy
  3. Memory - Medium
  4. Memory - Hard

    Arithmetic

  5. 2 Pails Puzzle
  6. 3 Jugs Puzzle
  7. 3 Jugs Puzzle in Barycentric Coordinates
  8. 3-Term Arithmetic Progression
  9. Abacus in Various Number Systems
  10. Algorithm for Computing LCM
  11. Aliquot Game
  12. Arithmetic Operations in Various Bases
  13. Broken Calculator
  14. Coin Game
  15. Collatz Conjecture
  16. Counting and Grouping
  17. Distributive Law
  18. EAN-13 Encoding
  19. Equivalent Fractions
  20. Even, Odd and Total Number of Digits
  21. Euclid's Game
  22. Farey Fractions
  23. Flipping Items Simultaneously
  24. Fraction Comparison
  25. GCD(M, N) × LCM(M, N) = M × N
  26. Geometric view of the Sieve of Eratosthenes
  27. ISBN Encoding
  28. Kaprekar's Iterations and Numbers
  29. Lattice Multiplication
  30. Long Multiplication
  31. Less, Equal, More
  32. Make An Identity
  33. Napier Bones
  34. Plain Multiplication table up to 20×20
  35. Product of Simple Fractions
  36. Sieve of Eratosthenes
  37. Sieve of Squares
  38. Soroban in Various Number Systems
  39. Suan Pan in Various Number Systems
  40. Sums and Products
  41. Sums of Powers of Digits
  42. Two Simple Equations
  43. Universal Product Code
  44. Zeros and Nines

    Algebra

  45. A Line in a Square Grid
  46. Averages, Arithmetic and Harmonic Means
  47. Binary Color Device
  48. Breaking Chocolate Bars
  49. Candy Game: Integer Iterations on a Circle
  50. Chebyshev polynomials
  51. Complex Numbers: A Dynamic Tool
  52. Constructible Numbers
  53. Counting Chips On a Circle
  54. Counting Squares in a Square
  55. Counting Triangles
  56. Counting Triangles II
  57. Divisibility Problem
  58. Dots and Fractions
  59. Ducci Sequences
  60. Egyptian Multiplication
  61. First Geometric Interpretation of Complex Numbers
  62. Four Pegs That Form a Square
  63. Frieze Patterns
  64. Geometric Construction of Roots of Quadratic Equation
  65. Graph and Roots of a Third Degree Polynomial
  66. Graph and Roots of Quadratic Polynomial
  67. Heads and Tails (counting, invariance)
  68. Identities in the Multiplication Table
  69. Integer Iterations on a Circle II (superposition principle)
  70. Integer Iterations on a Circle III
  71. Integers and Rectangles
  72. Integers and Rectangles: a Proof by Induction
  73. Integers and Rectangles: Two Simple Proofs
  74. Interest Calculations
  75. Josephus Flavius Game
  76. Langford and Skolem Sequences
  77. Letter Count in a Sentence
  78. Lights Out
  79. Logistic Model
  80. Loop or Halt
  81. Mean Values
  82. Merlin's Magic Squares (modular arithmetic, boolean and linear algebra)
  83. Mini Lights Out
  84. Minimax Principle
  85. Minimax Theorem
  86. Modular Arithmetic
  87. Nine Squares
  88. Parrondo Paradox Via Simpson Paradox
  89. Peasant Multiplication
  90. Pennies in Boxes
  91. Plus or Minus (parity, invariance)
  92. PolygonalNumbers
  93. Proizvolov's Identity
  94. Pythagorean Triples
  95. Pythagorean Triples Calculator
  96. Quasigroup on a Triangular Grid
  97. Rectangle on a Chessboard
  98. Right Replacement
  99. Self-descriptive Strings
  100. Self-documenting Sentences
  101. Sierpinski's Gasket and Dihedral Symmetry
  102. Signs and Sums in a Sequence
  103. Signs in a Matrix
  104. Simple Cellular Automaton
  105. Simpson Paradox
  106. Sine of a Sum Formula
  107. Solitaire in Two Colors
  108. Squares and Circles (parity, invariance)
  109. Squares, Circles, and Triangles (modular arithmetic, invariance)
  110. Steinhaus' Problem
  111. Sum of Consecutive Integers is Triangular
  112. Sum of Consecutive Odd Numbers is Square
  113. Sum of Consecutive Triangular Numbers is Square
  114. Toads And Frogs Puzzle (counting, logic, problem solving)
  115. Two Colors in Two Rows
  116. Two Colors in Two Rows II
  117. What's next?
  118. Word Problems

    Logic

  119. Lewis Carroll's Bilateral Diagrams
  120. Lewis Carroll's Trilateral Diagrams
  121. Lewis Carroll's Logic Game
  122. Venn Diagrams I
  123. Venn Diagrams II
  124. Formal Systems - an Example

    Fractals

  125. All Peano Curves
  126. Apollonian Gasket
  127. Color Cycling on the Mandelbrot Set
  128. Cantor sets
  129. Dot Patterns, Pascal Triangle and Lucas Theorem
  130. Emergence of Chaos
  131. Fractal Curves and Dimension
  132. Iterated Function Systems
  133. Iterations in the Mandelbrot Set
  134. Koch's Snowflake
  135. Mandelbrot Set and Indexing of Julia Sets
  136. Plane Filling Curves: Hilbert's and Moore's
  137. Plane Filling Curves: Peano's and Wunderlich's
  138. Sierpinski Gasket By Common Trema Removal
  139. Sierpinski Gasket Via Chaos Game
  140. Sierpinski Gasket by Trema Removal
  141. The Chaos Game: Address Space vs IFS
  142. Variations on the Theme of Tremas
  143. Variations on the Theme of Tremas II

    Probability

  144. Bertrand's Paradox
  145. Binomial Distribution
  146. Birds On a Wire
  147. Buffon's Noodle
  148. Chess Players Truel
  149. Conditional Probability and Independent Events
  150. Family Statistics
  151. Lewis Carroll's pillow problem
  152. Monty Hall Dilemma
  153. Parrondo Paradox
  154. Random Clock Hands
  155. Stick Broken Into Three Pieces (Cartesian Coordinates)
  156. Stick Broken Into Three Pieces (Trilinear Coordinates)
  157. Who Needs Monty?

    Visual Illusions

  158. Assimilation Illusion
  159. Balls and Shadows Illusion
  160. Brick Wall Illusion
  161. Bulging Lines illusion
  162. Circular Poggendorff Illusion
  163. Continuous Line Illusion
  164. Cube Counting Ambiguity
  165. Day's Sine Illusion
  166. Delboeuf Illusion
  167. Distorted Lines Illusion
  168. Divergent Spirals Illusion
  169. Gerbino's Illusion
  170. Heaver: A Visual Illusion
  171. Height/Width Interplay Illusion
  172. Hering Illusion
  173. Hermann Grid Illusion
  174. Importance of Occluders: Motion Binding
  175. Judd Illusion
  176. Kanizsa Triangle
  177. Müller-Lyer Illusion
  178. Necker Cube
  179. Occluded Wheel Illusion
  180. Orbison Illusion
  181. Parallelogram Illusion
  182. Perspective Illusion
  183. Poggendorff Illusion
  184. Poggendorff Illusion II
  185. Ponzo Illusion
  186. Ponzo Illusion II
  187. Reverse Spoke Illusion
  188. Revolving Circles Illusion
  189. Rotating Top Illusion
  190. Scintillating Grid Illusion
  191. Shepard's Parallelogram Illusion
  192. Shifting Cube Illusion
  193. Shimmer Illusion
  194. Simultaneous Contrast Illusion
  195. Simultaneous Contrast Illusion II
  196. Simultaneous Contrast: Koffka's Illusion
  197. Simultaneous Contrast: White's Illusion
  198. Simultaneous Contrast: White's Illusion, V-Variant
  199. Staircase Illusion
  200. Twisted Cord Illusion
  201. Unfolding Spirals Illusion
  202. Wundt Illusion
  203. Wundt Block Illusion
  204. Zöllner's Illusion

    Calculus

  205. Addition and Subtraction of Functions
  206. Cartesian Coordinate System
  207. Cavalieri's Principle
  208. Derivative of Sine and Cosine
  209. Distance From a Point to a Straight Line
  210. Equations of a Straight Line
  211. Estimating Circumference of a Circle
  212. Function, Derivative and Integral
  213. Graph of a Polynomial
  214. Graph of a Polynomial Defined by Its Roots
  215. Horner's Method
  216. Inflection Points of Fourth Degree Polynomials
  217. Lagrange Interpolation
  218. Maximum Volume of a Cut Off Box
  219. Mistrust Intuition of the Infinite
  220. Naturally Discontinuous Functions
  221. Newton's Method
  222. Riemann Sums - Function Integration
  223. Rolle's and Mean Value Theorems
  224. Taylor Series Approximation to Cosine
  225. Two Circles and a Limit

    Math Magic

  226. An Arithmetic Magic Trick
  227. Arithmetic magic matrix
  228. Bachet's Magic Trick
  229. Barcode Magic
  230. Calendar Magic
  231. Gergonne's Magic Trick
  232. Hummer's Mind Reader
  233. Magic in Square
  234. Math Telepathy
  235. Number Guessing Game
  236. Two Numbers Guessing Game

    Combinatorics

  237. 3-Colors Tower of Hanoi
  238. Affirmative Action Problem
  239. Balanced Incomplete Block Design on a Triangular Grid
  240. Bicolor Tower of Hanoi
  241. Blithe 12
  242. Cheapest Link & Kruskal's Algorithms - skill practice
  243. Chvatal's Art Gallery Theorem
  244. Coin change
  245. Coloring a Graph
  246. Critical Path in a Weighted Digraph
  247. Scheduling and Critical Path Algorithm
  248. de Bruijn Cycle
  249. Dominoes on a Chessboard
  250. Euler & Hamilton Paths - skill practice
  251. Five Lattice Points
  252. Fleury's Algorithm and Euler's Paths and Cycles
  253. Flipping Items Simultaneously II
  254. Flipping pancakes
  255. Gray Codes
  256. Group multiplication of permutations
  257. Finding Hamilton Paths and Circuits
  258. Hanoing
  259. Happy 8
  260. Hex Can't End in a Draw
  261. How to define a permutation
  262. Infinite Latin Square
  263. Jumping Isn't Everything
  264. Latin Squares
  265. Leibniz and Pascal Triangles
  266. Life-like Automaton With Definable Rules
  267. Light Bulbs in a Circle
  268. Listing All Combinations
  269. Listing All Permutations
  270. Listing All Permutations: Johnson-Trotter Method
  271. Lucky 7
  272. Make Your Move, Kid!
  273. Marriage Problem, simulation
  274. Mazes
  275. Minumum Spanning Trees and Kruskal's Algorithm
  276. Monotone Number Sequences
  277. Number sequences and Pigeonhole
  278. Party Acquaintances
  279. Permutations as a Product of Transpositions
  280. Product of Permutations
  281. Project Scheduling
  282. Property of the Powers of 2
  283. Recombining Permutations
  284. Sam Loyd's Fifteen
  285. Scrub Tile Puzzle
  286. Shuttle Puzzle
  287. Simple Graphs
  288. Simple Graph Practice
  289. Sliders (slider games on torus, Moebius strip, projective plane, permutations, cycles)
  290. Straight Tromino on a Chessboard
  291. The Game of Life
  292. Tool for Constructing Orthogonal Latin Squares
  293. Tower of Hanoi
  294. Tower of Hanoi, the Hard Way
  295. Traveling Salesman Problem
  296. Two Men of Tibet
  297. Universal Coloring
  298. Y Can't End in a Draw

    Combinatorial Games

  299. Date Game
  300. Dawson's Chess
  301. Dawson's Kayles
  302. Fraction Game
  303. Grundy's Game
  304. Hex 7
  305. Kayles
  306. Nim
  307. Nimble
  308. Northcott's game
  309. One Pile
  310. Plainim
  311. Plainim Misère
  312. Scoring
  313. Scoring Misère
  314. Silver Dollar Game
  315. Silver Dollar Game With No Silver Dollar
  316. Sticks, Jess realization of Scoring Misère
  317. Sticky Problem
  318. Sticky Problem II
  319. Subtraction Game
  320. TacTix
  321. Take-Away Games
  322. Turning Turtles
  323. Wythoff's Nim
  324. Wythoff's Nim II

    Puzzles & Games

  325. 3-Chain
  326. 4×4×4 Tic Tac Toe (spacial visualization)
  327. Achieve the Equilibrium
  328. Bulls and Cows
  329. Caterpillar
  330. Changing Colors (combinatorics, invariance)
  331. Color Balls
  332. Computer Mastermind (logic and problem solving)
  333. Eight Digits Puzzle
  334. Farmer and Wife To Catch Rooster and Hen (invariance, city block distance)
  335. Fif (multiple representations)
  336. Fill a Hexagonal Grid (space visualization)
  337. Fish Soup Game (multiple representations)
  338. Four Knights (multiple representations)
  339. Full House
  340. Goat, Cabbage and Wolf to move across a river (multiple representations)
  341. Hare and Hounds
  342. Jam (multiple representations)
  343. Looping Chips (multiple representations)
  344. Mastermind (logic and problem solving)
  345. Mastermind Variants (logic and problem solving)
  346. Mirror Maze
  347. Peg Solitaire (logic, problem solving, subproblems)
  348. Pick A Label (logic, problem solving)
  349. Reverse Solitaire (logic, problem solving, subproblems)
  350. Sending scouts into the desert (algebra, geometric series, problem solving)
  351. Seven Coins Puzzle (multiple representations)
  352. Shuttle Puzzle Practice
  353. Solitaire on a Circle (iterations, invariance)
  354. Stones in Cups
  355. Super Nim
  356. Thugs and Bowmen
  357. Tiling a Triangulated Hexagon (a proof without words)
  358. Toads And Frogs in 2 dimensions (repeated application of same idea)
  359. Tromino Puzzle
  360. Vector Addition and Subtraction
  361. Vowel Movement
  362. Wolf and Rabbit

    Geometric Fallacies

  363. A Circle With Two Centers
  364. All Triangles Are Isosceles
  365. Curry's Paradox
  366. Delian Problem Solved
  367. A Faulty Dissection
  368. Hooper's Paradox
  369. Is Every Trapezoid Parallelogram?
  370. John Sharp's Paradox
  371. Langman's Paradox
  372. Popping A Square
  373. Rouse Ball's Fallacy
  374. Sam Loyd's Son's Dissection
  375. Two Perpendiculars From a Point to a Line

    Social Science

  376. Adams' Apportionment Method
  377. Banzhaf Power Index Calculator
  378. Fair Division: Method of Lone Divider
  379. Fair Division: Method of Markers
  380. Fair Division: Method of Sealed Bids
  381. Fair Division: Method of Sealed Bids II
  382. Fast Power Indices
  383. Five Methods of Apportionment
  384. Four Voting Methods
  385. Hamilton's Apportionment Method
  386. Huntington-Hill Apportionment Method
  387. Jefferson's Apportionment Method
  388. Method of Markers II
  389. Shapley-Shubik Power Index Calculator
  390. Social Choice
  391. Webster's Apportionment Method
  392. Weighted Voting and Power Indices

    Elementary Geometry

  393. A Property of Equiangular Polygons
  394. Al-Nayrizi's Construction
  395. Angle: An Illustrated Classification
  396. Angle Bisector
  397. Angle Bisectors in Ellipse
  398. Angle Bisectors in Ellipse II
  399. Angle Bisectors On Circumcircle
  400. Angle Subtended by a Diameter
  401. Archimedes' Law of the Lever
  402. Area of Median Triangle
  403. Area of Parallelogram
  404. Area of the Union of Two Squares
  405. Area of Triangle
  406. Arithmetic in Disguise
  407. Between Major and Minor Circles
  408. Building a Bridge
  409. Carpet With a Hole
  410. Circle and Hyperbola as Lighthouse Curves II
  411. Circles and Semicircles in Rectangle
  412. Circumcenter on Angle Bisector
  413. Classification of Quadrilaterals
  414. Common Chord and a Tangent
  415. Conic Sections as Loci of Points
  416. Construction of the Angle Bisector
  417. Construction of the Perpendicular Bisector
  418. Diagonal Count
  419. Dissection of Cyclic Quadrilateral
  420. Divide a Circle into N Parts of Equal Area
  421. E. W. Dijkstra's Proof of the Pythagorean Theorem
  422. Eight Equal Tangents
  423. Equal Circles, Medial Triangle and Orthocenter
  424. Equidecomposition of a Rectangle and a Square
  425. Equidecomposition of Two Rectangles
  426. Equidecomposition of a Triangle and a Rectangle
  427. Equidecomposition of a Triangle and a Rectangle II
  428. Equidecomposition of Two Parallelograms
  429. Equilic Quadrilateral I
  430. Equilic Quadrilateral I, A Variation
  431. Equilic Quadrilateral II
  432. Euclid's Segment Division
  433. Existence of the Circumcenter
  434. Fibonacci Bamboozlement
  435. Famous Theorem
  436. Finding Circle Center By Ruler Alone
  437. Focal Definition of Ellipse
  438. Focus and Directrix of Ellipse
  439. Four Hinged Squares
  440. Geoboard
  441. Harmonic Mean Sangaku
  442. Hippocrates' Squaring of a Lune
  443. Hunting Right Angles
  444. Huygens' proof of the Pythagorean Theorem
  445. Inscribed and Central Angles in a Circle
  446. Inscribed Angles
  447. Inscriptible Quadrilateral: An Illustration
  448. Lean Napoleon's Triangles
  449. Maimed Cake
  450. Mathematics in Pizzeria
  451. Napoleon's Theorem via Two Rotations
  452. Napoleon on Hinges
  453. Octagon in Parallelogram
  454. Octagon in Parallelogram: Overlapping Regions
  455. Olof Hanner's Jigsaw Puzzle
  456. Orthocenters of Two Triangles Sharing Circumcenter and Base
  457. Pappus' Generalization of Euclid I.47
  458. Orthodiagonal and Cyclic Quadrilaterals
  459. Parallel Chords II
  460. Parallel Lines in a Quadrilateral
  461. Pascal: Necessary and Sufficient
  462. Polygon Metamorphosis
  463. Property of Angle Bisectors
  464. Property of Rhombi
  465. Pythagorean Theorem By Dissection
  466. Pythagorean Theorem By Dissection II
  467. Pythagorean Theorem by Hinged Dissection
  468. Pythagorean Theorem by Hinged Dissection I
  469. Pythagorean Theorem by Hinged Dissection II
  470. Pythagorean Theorem: J. Adams' Proof by Dissection
  471. Pythagorean Theorem: Versluys' Proof by Dissection
  472. Pythagoras' Theorem By Sheer Shearing
  473. Pythagoras' Theorem By Similarity
  474. Pythagoras' Theorem Solely By Sliding
  475. Pythagorean Segment Division
  476. Quadrilateral With Equal Opposite Sides And Angles
  477. Rhombus in a Cyclic Quadrilateral
  478. Secant Angles
  479. Secant Angles II
  480. Shearing a Polygon into a Triangle of Equal Area
  481. Simple Quadrilaterals Tessellate the Plane
  482. Shearing and Translation in Pythagorean Pants
  483. Square From Nowhere
  484. Square in Chair
  485. Squaring a Rectangle
  486. Three Touching Circles
  487. Translation Transform
  488. Triangle Areas in a Parallelogram
  489. Triangle Areas in a Parallelogram II
  490. Triangle Classification
  491. Trigonometric Functions
  492. Two Color Coloring of the Plane
  493. Two Intersecting Circles
  494. Two Parallels in a Triangle and One More
  495. Two Symmetric Triangles Are Directly Equidecomposable
  496. Two Symmetric Triangles Are Directly Equidecomposable II
  497. Two Symmetric Triangles Are Directly Equidecomposable III
  498. Two Symmetric Triangles Are Directly Equidecomposable IV
  499. Two Touching Circles
  500. Van Schooten's and Pompeiu's Theorems
  501. Viviani's Theorem
  502. Viviani's Theorem II

    Stereometry

  503. Angles in a Cube I
  504. Angles in a Cube II
  505. Angles in a Cube III
  506. Bimedians in a Regular Tetrahedron
  507. Cut the Cone
  508. Cut the Cube
  509. Cut the Cylinder
  510. Regular Polyhedra - Platonic Solids
  511. Möbius' Strip
  512. Stereographic Projection
  513. Three Pyramids are Better Than Two

    Outline (Fill-It-In) Mathematics

  514. A bunch of fill-it-in outline problems and solutions

 

    Geometry

  1. 120° Breeds 90°
  2. 2N-Wing Butterfly Theorem
  3. 3 Isosceles Trapezoids
  4. 5-Star and A Circle
  5. 6 to 9 Point Circle
  6. 60o Breeds 90o
  7. 9-point Circle as a Locus of Concurrency
  8. A Circle Related to Incenter and Circumcenter
  9. A Line in Triangle Through the Circumcenter
  10. A Parallelogram in Triangle
  11. A Sangaku Follow-Up on an Archimedes' Lemma
  12. A Sangaku: Two Unrelated Circles
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Stronger Triangle Inequality
  16. A Triangle of Antreas Hatzipolakis
  17. Adams' Circle
  18. Addition of Radius-Vectors
  19. Altitudes and the Power of a Point
  20. An Isoperimetric Theorem
  21. Angle Bisector in Parallelogram
  22. Angle Bisector in Touching Circles
  23. Angle Bisectors and Perpendiculars in a Quadrilateral
  24. Angle Bisectors in a Quadrilateral
  25. Angle Preservation Property
  26. Angle Trisection by Archimedes
  27. Angle Trisection by Hippocrates
  28. Angle Trisection by Paul Vjecsner
  29. Angle Trisectors on Circumcircle
  30. Angles in Triangle Add to 180o
  31. Another Pair of Twins in an Arbelos
  32. Anticomplementary Triangle Surprise
  33. Antiparallel
  34. Apollonian Circles Theorem
  35. Apollonius Problem
  36. Application of Ceva's Theorem
  37. Archimedes' Quadruplets
  38. Archimedes Triangle and Squaring of Parabola
  39. Archimedes' Twin Circles and a Brother
  40. Area of an Arbelos
  41. Areas and Centroid in a Triangle
  42. Areas in Triangle
  43. Areas in Triangle II
  44. Arithmetic Mean Sangaku
  45. Arithmetic Mean Sangaku II
  46. Around the Incircle
  47. Around the Incircle in a Polygon
  48. Asymmetric Propeller I
  49. Asymmetric Propeller II
  50. Asymmetric Propeller III
  51. Barycentric Coordinates: a Tool
  52. Bevan's Point and Theorem
  53. Bisectal Circle
  54. Book of Lemmas: Proposition 1
  55. Book of Lemmas: Proposition 2
  56. Book of Lemmas: Proposition 3
  57. Book of Lemmas: Proposition 5
  58. Book of Lemmas: Proposition 6
  59. Book of Lemmas: Proposition 9
  60. Book of Lemmas: Proposition 10
  61. Book of Lemmas: Proposition 12
  62. Book of Lemmas: Proposition 13
  63. Böttcher's Proof of the Pythagorean Theorem
  64. Böttcher's Proof Variant
  65. Bottema Shatters Japan's Seclusion
  66. Bottema's Theorem
  67. Bottles in a Slanted Rack
  68. Bottles in a Wine Rack
  69. Bottles in a Wine Rack: Proofs and Generalizations
  70. Brahmagupta's Theorem
  71. Brianchon in Ellipse
  72. Brianchon's Theorem
  73. Bride's Chair
  74. Broken Chord Theorem
  75. Broken Chord Theorem by Paper Folding
  76. Broken Chord Theorem: G. Patruno's proof
  77. Broken Chord Theorem: S. Anderson's proof
  78. Butterfly in Ellipse
  79. Butterfly Problem
  80. Cabart's Collinearity
  81. Cantor's Theorem
  82. Cardioid As Envelope of Circles
  83. Cardioid As Epicycloid
  84. Carnot's Theorem
  85. Carpets in a Parallelogram
  86. Carpets in a Quadrilateral
  87. Carpets in a Quadrilateral II
  88. Carpets Theorem
  89. Carpets Theorem With Parallelograms
  90. Center Circles and Their Chains
  91. Centroids in a Polygon
  92. Ceva and Menelaus Theorems for Angle Bisectors
  93. Ceva's Theorem
  94. Ceva's Theorem (PWW)
  95. Cevian Nest
  96. Cevian Triangle
  97. Chain of Inscribed Circles
  98. Chasing Angles in Pascal's Hexagon
  99. Chasles' Theorem, a Proof
  100. Cherchez le quadrilatère cyclique
  101. Cherchez le quadrilatère cyclique II
  102. Chords, Concurrency and Orthic Triangle
  103. Chords in Cardioid
  104. Circle and Hyperbola as Lighthouse Curves
  105. Circle Chains and Inscribed Angles
  106. Circle Concurrency and Spiral Similarity
  107. Circle Crosses Triangle
  108. Circle Division by Euclid and Leonardo
  109. Circle Inscribed in a Circular Segment
  110. Circle of Similitude
  111. Circle-Stacking Theorem
  112. Circles And Parallels
  113. Circles and Reflections in a Triangle
  114. Circles in Morley's Triangles
  115. Circles Lined on the Legs of a Right Triangle
  116. Circles On Cevians
  117. Circles Tangent to Circumcircle
  118. Circles through the Orthocenter
  119. Circles with Tangent Diameters
  120. Circumcevian Triangle
  121. Cleavance Center
  122. Cleaver
  123. Clifford Chain
  124. Clifford's Lemma
  125. Coaxal Circles Theorem
  126. Colburn's Proof of the Pythagorean Theorem
  127. Common Tangents to Two Circles I
  128. Common Tangents to Two Circles II
  129. Complete Quadrilateral
  130. Concyclic Points in Equilateral Bumps
  131. Concurrency in Analytic Geometry
  132. Concurrency in Arbelos
  133. Concurrency in the Intouch Triangle
  134. Concyclic Circumcenters: A Dynamic View
  135. Concyclic Points in a Triangle
  136. Conjugate Diameters in Ellipse
  137. Construction of Conics from Pascal's Theorem
  138. Consecutive Isosceles Decomposition
  139. Conway's Circle
  140. Copernicus' Theorem
  141. Crossed-Lines Construction of Shapes of Constant Width
  142. Cyclic Hexagon
  143. Cyclic Incenters
  144. Cycloids
  145. Diameters and Chords
  146. Directly Similar Figures
  147. Desargues' Theorem
  148. Discovery of Duane DeTemple
  149. Dividing a Segment into N parts: Besteman's Construction
  150. Dividing a Segment into N parts: Besteman's Construction II
  151. Dividing a Segment into N parts: GLaD Construction
  152. Dividing a Segment into N parts: SaRD Construction
  153. Dividing Evenly a Quadrilateral
  154. Dividing Evenly a Quadrilateral II
  155. Double Generation Theorem
  156. Douglas' Theorem (complex variables, linear algebra)
  157. Droz-Farny Circles
  158. Droz-Farny Line Theorem
  159. Dual to Pappus' Theorem
  160. Dynamic Construction of Ellipse and Other Curves
  161. Easy Construction of Bicentric Quadrilateral
  162. Easy Construction of Bicentric Quadrilateral II
  163. Eight Point Circle
  164. Ellipse, how to draw
  165. Ellipse in Arbelos
  166. Ellipse Between Two Circles
  167. Equal Angles in Two Circles
  168. Equal Angles in Two Circles II
  169. Equal Areas In Parallelogram à la Pythagoras
  170. Equal Incircles Theorem
  171. Equichordal Curves
  172. Equilateral Triangle, Straight Line and Tangent Circles
  173. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  174. Equilateral Triangles on Segments of Equilic Quadrilateral
  175. Equilateral Triangles On Sides of a Parallelogram
  176. Equilateral Triangles On Sides of a Parallelogram II
  177. Equilateral Triangles on Sides of a Quadrilateral
  178. Escher's Theorem
  179. Euler Line Cuts Off Equilateral Triangle
  180. Euclid I.43
  181. Euclid I.43 Extended
  182. Euclidean Construction of Center of Ellipse
  183. Euclidean Construction of Tangent to Ellipse
  184. Extouch Triangle in Poncelet Porism
  185. Eyeball Theorem
  186. External Angle Bisector
  187. Fagnano's problem
  188. Fagnano's problem II
  189. Fagnano's Problem, Third Solution
  190. Fagnano's Problem, Fourth Solution
  191. Fagnano's Problem in Reverse
  192. Fermat point and 9-point Centers
  193. Fermat's Point
  194. Fermat Points and Concurrent Euler Lines I
  195. Fermat Points and Concurrent Euler Lines II
  196. Feuerbach's theorem
  197. Feuerbach's Theorem: A Proof
  198. First Vecten Point
  199. Five Cardioids
  200. Five Concyclic Points
  201. Five Incircles in a Square
  202. Five Incircles Theorem
  203. Fixed Point of Circles Orthogonal to the Given One
  204. Fixed point on a rod rotating around a triangle
  205. Focal Properties of Parabola
  206. Foot of Altitude and Minimum Distance
  207. Four Centroids and Parallels
  208. Four 9-Point Circles in a Quadrilateral
  209. Four Circles In a Triangle
  210. Four Concurrent Lines in a Cyclic Quadrilateral
  211. Four Incident Circles
  212. Four Incircles in Equilateral Triangle
  213. Four Pedal Circles
  214. Four Touching Circles
  215. Four Touching Circles II
  216. Four Touching Circles III
  217. Four Triangles, One Circle
  218. Four Turtles (or Bugs, or Dogs, etc.)
  219. Friendly Kiepert's Perspectors
  220. From Foci to a Tangent in Ellipse
  221. Fundamental Theorem of 3-Bar Motion
  222. Fuss' Theorem
  223. Generalizing a Romanian Olympiad Problem
  224. Geometric Optimization from the Asian Pacific Mathematical Olympiad
  225. Gergonne and Medial Triangles Are Orthologic
  226. Gergonne and Soddy Lines Are Perpendicular
  227. Gergonne in Ellipse
  228. Glide Reflection
  229. Golden Ratio in an Irregular Pentagon, Construction I
  230. Golden Ratio in an Irregular Pentagon, Construction II
  231. Golomb's inductive proof of a tromino theorem
  232. Gothic Arc
  233. Grebe: from Ladies' Diary to Carroll's Pillow
  234. Grebe's Theorem
  235. Greg Markowsky's Problem for Parabola
  236. Griffiths Points in Cyclic Quadrilateral
  237. Harmonic Ratio
  238. Hart Circle
  239. Hart's Inversor
  240. Haruki's Lemma
  241. Heron's Problem
  242. Hexagon Parallel to Medial Triangle
  243. Hexagon Parallel to Orthic Triangle
  244. Hinged Squares
  245. Hjelmslev Theorem
  246. Hofstadter Triangles and Points
  247. Homologous Lines under Three Spiral Similarities
  248. Homothety
  249. Hypocycloid Families
  250. Ian McGee's Observation
  251. Illustration of Pascal's Theorem
  252. Incenters in Cyclic Quadrilateral
  253. Incidence in Feuerbach's Theorem
  254. Incident Center Circles and Cardioid
  255. Incircles, Circumcircles and Inscribed Angles
  256. Inscriptible and Exscriptible Quadrilaterals
  257. Internal Tangents to Three Circles
  258. Intouch Triangle in Poncelet Porism
  259. Invariant Area Sweep Implies Pythagorean Theorem
  260. Inversion: Reflection in a Circle
  261. Inversion in the Incircle
  262. Inversion Tool
  263. Inversion with Negative Power
  264. Isogonal and Isotomic Conjugalities
  265. Isogonal Concurrency
  266. Isogonal image of the circumcircle
  267. Isosceles on the Sides of a Triangle
  268. Isotomy and Isogonality Hand-in-Hand
  269. Iterations in Geometry, an example
  270. Iterations in Geometry, Generalization
  271. Iterations on a Circle Through Three Points
  272. Iterations on Euler Lines
  273. Kiepert's Theorem
  274. Kürschak's Tile and Theorem
  275. Law of Cosines
  276. Law of Cosines: PWW
  277. Law of Cosines: An Unfolded Variant
  278. La Hire's Theorem
  279. La Hire's Theorem, a Variant
  280. La Hire's Theorem in Ellipse
  281. Lighthouse Curves
  282. Locus of Points in a Given Ratio to Two Points
  283. MacLaurin's Construction of Conics
  284. Malfatti's Problem
  285. Malfatti's Problem, Hart's Solution
  286. Maxwell's Theorem
  287. Median in Touching Circles
  288. Medians in a Quadrilateral
  289. Menelaus and Ceva Theorems
  290. Menelaus From Ceva
  291. Midline in Similar Triangles
  292. Midpoints from Gergonne Triangle
  293. Midpoints of the Lines Joining In- and Excenters
  294. Minkowski's addition of convex shapes
  295. Miquel's Point
  296. Miquel's Point of a 4-line Via Spiral Similarity
  297. Miquel's Theorem for Circles
  298. Mirror on the Wall
  299. Mirror property of the altitudes
  300. Mirror Property of Altitudes via Pascal's Hexagram
  301. Mixtilinear Circles and Concurrence
  302. Monge via Desargues
  303. Monge via Desargues II
  304. More Bottles in a Wine Rack
  305. Morley Constellation
  306. Morley's Theorem
  307. Morley's Theorem, Cardioids and Such
  308. Morley's Theorem, G. Zsolt Kiss' proof
  309. Nagel Line
  310. Nagel point
  311. Nagel Point of Medial Triangle
  312. Nagel's Theorem
  313. Napoleon-Barlotti Theorem
  314. Napoleon Theorem by Plane Tessellation
  315. Napoleon's Relatives
  316. Napoleon's Theorem by Transformation
  317. Neuberg Sangaku
  318. Newton's Construction of Conics
  319. Newton's Theorem
  320. Nine Point Circle
  321. No Equilateral Triangles, Please
  322. Nobbs' Points, Gergonne Line
  323. Non-classic Way To Draw Cardioid
  324. No-Pedal Collinearity
  325. On Bottema's Shoulders
  326. On Bottema's Shoulders II
  327. One Sided Surface in 4D
  328. Optical Property of Ellipse
  329. Optimization Problem in Acute Angle
  330. Orthocenter and Three Equal Circles
  331. Orthocentric System From Rectangle
  332. Orthogonality in Isosceles Triangles
  333. Orthologic Triangles in a Quadrilateral
  334. Orthopole
  335. Pairs of Homologous Lines under Spiral Similarities
  336. Pairs of Incircles in a Quadrilateral
  337. Pappus' Theorem
  338. Parabola As Envelope of Straight Lines
  339. Parabola As Envelope II
  340. Parabolic Mirror
  341. Paragon
  342. Parahexagon
  343. Parallel Chords
  344. Parallel Chords in Ellipse
  345. Parallel Lines and Similar Triangles
  346. Parallelogram and Similar Triangles
  347. Parallelogram Law
  348. Parallelogram Law: A PWW
  349. Parallelogram Law: A Tessellation
  350. Pascal in a Cyclic Quadrilateral
  351. Pascal Lines: Steiner and Kirkman Theorems
  352. Pascal Lines: Steiner and Kirkman Theorems II
  353. Pascal's Theorem
  354. Peaucellier Linkage
  355. Pedal Collinearities
  356. Pedal Polygons
  357. Pedal Triangle and Isogonal Conjugacy
  358. Pedoe's Theorem
  359. Perigal's Proof of the Pythagorean Proposition
  360. Perpendicular Bisectors in an Inscriptible Quadrilateral
  361. Perpendicular Bisectors in an Inscriptible Quadrilateral II
  362. Pinning Butterfly on Radical Axes
  363. Pivot Theorem
  364. Point common to two similar rectangles
  365. Polar Circle
  366. Pole and Polar with respect to a Triangle
  367. Poles and Polars
  368. Poles, Polars and Quadrilaterals
  369. Polygon from midpoints of its sides
  370. Polygons Formed by Perpendicular Bisectors
  371. Poncelet's Porism
  372. Poncelet's Porism in Ellipses
  373. Poncelet's Theorem
  374. Problem in an Equilateral Triangle
  375. Problem in an Equilateral Triangle II
  376. Problem In an Isosceles Right Triangle
  377. Problem in Three Squares
  378. Problem of Equal Steps
  379. Problem of Equal Steps II
  380. Problem of polygon construction
  381. Problem with Equilateral Triangles
  382. Product of Rotations
  383. Projections on Internal and External Angle Bisectors
  384. Projective Collinearity in a Quadrilateral
  385. Projective Generalization of Maxwell's Theorem
  386. Projective Proof of Pascal's Theorem
  387. Proof Without Words for the Pythagorean Theorem
  388. Properties of the Circle of Similitude
  389. Property of 6-Parpolygon
  390. Property of Angle Bisectors II
  391. Property of Circumscribed Quadrilaterals
  392. Property of Points Where In- and Excircles Touch a Triangle
  393. Property of the Line IO
  394. Property of the Line IO II
  395. Property of the Line IO: a proof from the Book

    Ptolemy on Hinges

  396. Pythagoras and Vecten Break Japan's Isolation
  397. Pythagorean Condition in An Isosceles Right Triangle
  398. Pythagorean Theorem: Subtle Dangers of Visual Proof
  399. Pythagorean Theorem By Euclid
  400. Pythagorean Theorem By Plane Tessellation
  401. Pythagorean Theorem By Hexagonal Tessellation
  402. Pythagorean Theorem By Rearrangement
  403. Pythagorean Theorem By Shearing and Rotations I
  404. Pythagorean Theorem By Shearing and Rotations II
  405. Pythagorean Theorem in an Orthodiagonal Quadrilateral
  406. Quadrilaterals Formed by Perpendicular Bisectors
  407. Quest for Paragon
  408. MathPro Logo
  409. Radical Axis and Center, an Application
  410. Radical Axis of Circles Inscribed in a Circular Segment
  411. Radical Center
  412. Radius and Construction of a Mixtilinear Circle
  413. Radius of a Circle by Paper Folding
  414. Rectangle in Arbelos
  415. Reflection In Line
  416. Reflections in Ellipse
  417. Reflections of a Line Through the Orthocenter
  418. Reflections of a Point on the Circumcircle
  419. Reflections of the Orthocenter
  420. Remarkable Line in a Quadrilateral
  421. Remarkable Line in Cyclic Quadrilateral
  422. Rendezvous
  423. Right Isosceles Triangles on Sides of a Quadrilateral
  424. Rotation Transform
  425. Rotations in Disguise
  426. Rusty Compass Construction of Equilateral Triangle
  427. Salinon: from Archimedes Book of Lemmas
  428. Sangaku Iterations, Is it Wasan?
  429. Sangakus with a Mixtilinear Circle
  430. Sangaku with Three Mixtilinear Circles
  431. Seven Circles Theorem
  432. Squares in Arbelos
  433. Secant, Tangents and Orthogonality
  434. See-Saw Lemma
  435. Segment Trisection
  436. Shape of Constant Width
  437. Shearing Butterflies in Quadrilaterals
  438. Siamese Triangles
  439. Siamese Triangles II
  440. Similar Triangles on Sides of a Quadrilateral
  441. Similar Triangles on Sides and Diagonals of a Quadrilateral
  442. Simson Line
  443. Simson Line From Isogonal Perspective
  444. Simson Lines Of Diametrically Opposite Points
  445. Simsons and 9-Point Circles in Cyclic Quadrilateral
  446. sin(N)
  447. Six Circles Theorem (Elkies)
  448. Six Circles Theorem (Evelyn)
  449. Six Concyclic Points
  450. Six Concyclic Points II
  451. Six Incircles in an Equilateral Triangle
  452. Six Point Circle
  453. SixPoints, ThreeLines
  454. Soddy Circles and David Eppstein's Centers
  455. Some Properties of Napoleon's Configuration
  456. Sperner's Lemma
  457. Spira Mirabilis - Wonderful Spiral
  458. Spiral Similarity
  459. Square in a Right Triangle
  460. Square Inscribed in a Triangle
  461. Square Inscribed in a Triangle II
  462. Square Inscribed In a Triangle III
  463. Squares in Circles
  464. Squares on Sides of a Quadrilateral
  465. Squinting Eyes Theorem
  466. Star Construction of Shapes of Constant Width
  467. Steiner's Chain
  468. Steiner's porism
  469. Straight Edge Construction of Polar
  470. Symmedian and Antiparallel
  471. Symmedian and 2 Antiparallels
  472. Symmedian and the Tangents
  473. Symmedian in a Right Triangle
  474. Tangency Point of Two Circles
  475. Tangent and Secant
  476. Tangent Circles and an Isosceles Triangle Tangent Circles and an Isosceles Triangle II
  477. Tangent Circles in a Parallelogram
  478. Tangent in Concurrency
  479. Tangents and Diagonals in Cyclic Quadrilateral
  480. Tangents, Perpendiculars and Geometric Mean
  481. Tesseract
  482. Thébault's Problem I
  483. Thébault's Problem II
  484. Thébault's Problem III
  485. Thébault's Problem IV
  486. Thébault's Problem III, Proof
  487. Theorem of Complete Quadrilateral
  488. Theorems of Ceva and Menelaus, an Illustrated Generalization
  489. Three Concurrent Circles
  490. Three Congruent Circles by Reflection
  491. Three Congruent Circles by Reflection II
  492. Three Congruent Circles by Reflection III
  493. Three Incircles In a Right Triangle
  494. Three Incircles in a Triangle
  495. Three Isosceles Triangles
  496. Three Parabola Tangents
  497. Three Parallels in a Triangle
  498. Three Points on a Parabola
  499. Three Roads, Three Travelers
  500. Three Similar Polygons
  501. Three Similar Triangles
  502. Three Similar Triangles (but differently oriented)
  503. Three Similar Triangles (but differently oriented) II
  504. Three Squares and Two Ellipses
  505. Three Tangent Circles
  506. Three Tangents Theorem
  507. Three Tangents Theorem II
  508. Three Tangents, Three Chords in Ellipse
  509. Three Tangents, Three Secants
  510. Toothpick Construction of a Square
  511. Touching Circles
  512. Touching Circles and Concurrency
  513. Triangle and Segment Count in a Polygon
  514. Triangle Geometry
  515. Triangle Sliding on Two Straight Lines
  516. Triangles, Squares and Areas from Temple Geometry
  517. Triangles with Equal Area
  518. Triangular Billiard
  519. Trigonometric Form of Ceva's Theorem (a tool for discovery of trigonometric identities and other problems)
  520. Tromino as a Rep-tile
  521. Tucker Circles
  522. Tucker Circles Through Homothety
  523. Twin Segments in Arbelos
  524. Two Butterflies Theorem
  525. Two Butterflies Theorem II
  526. Two Butterflies Theorem III
  527. Two Circles in a Parallelogram
  528. Two Circles Inscribed in a Parallelogram
  529. Two Circles in a Square
  530. Two Circles in a Square II
  531. Two Circles in an Angle
  532. Two Circles on a Side of a Triangle
  533. Two Circumcircles And Two Pairs of Parallels
  534. Two Circumcircles in Triangle
  535. Two Congruent Circles by Reflection
  536. Two Homotheties in a Parallelogram
  537. Two Proofs of MenelausTheorem
  538. Two Quadrilaterals
  539. Two Quadruplets of Tangent Circles
  540. Two Rectangles in a Rectangle
  541. Two Simsons in a Triangle
  542. Two Squares and Another Square
  543. Two Triples of Similar Triangles
  544. Two Tangents to Parabola
  545. Two Triangles With Common Base and Altitude
  546. Varignon Parallelogram
  547. Vecten's Collinearity
  548. Vecten's Mesh
  549. Vecten's Theorem
  550. Versatile Theorem
  551. Weitzenböck Inequality
  552. What does one see inside a spherical mirror?
  553. When a Triangle is Equilateral
  554. Wittenbauer's Parallelogram
  555. Y. Sawayama's Lemma
  556. Zaslavsky's theorem

    Miscellaneous Demonstrations

  557. Calculation of the Digits of pi by the Spigot Algorithm
  558. Calculation of the Digits of pi (Faster Version)
  559. Convex Polygon Is the Intersection of Half Planes
  560. Disappearing Lines
  561. Disappearing Rectangles
  562. Distance Between Strings
  563. Experimentation with Dynamic Geometry Software: An Example
  564. Genetic Algorithm Solves the Toads and Frogs Puzzle
  565. Impossible Fork
  566. Impossible Frame
  567. Morphing
  568. Possible or Impossible?
  569. Projectile Trajectory
  570. Random Dot Stereograms
  571. Ratchet Effect
  572. Rotating Gears
  573. Sorting Algorithms
  574. Structural Constellation


Copyright © 1996-2009 Alexander Bogomolny

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