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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. Compare Fractions: Interactive Practice
  17. Counting and Grouping
  18. Distributive Law
  19. EAN-13 Encoding
  20. Equivalent Fractions
  21. Euclid's Algorithm: An Interactive Illustration
  22. Euclid's Game
  23. Even, Odd and Total Number of Digits
  24. Factoring with the Factor Tree
  25. Farey Fractions
  26. Flipping Items Simultaneously
  27. Fraction Comparison
  28. GCD and LCM via Factoring
  29. GCD(M, N) × LCM(M, N) = M × N
  30. Geometric view of the Sieve of Eratosthenes
  31. ISBN Encoding
  32. Kaprekar's Iterations and Numbers
  33. Lattice Multiplication
  34. Long Multiplication
  35. Less, Equal, More
  36. Make An Identity
  37. Make It All Zeros
  38. Napier Bones
  39. Plain Multiplication table up to 20×20
  40. Product of Simple Fractions
  41. Sieve of Eratosthenes
  42. Sieve of Squares
  43. Soroban in Various Number Systems
  44. Suan Pan in Various Number Systems
  45. Sums and Products
  46. Sums of Powers of Digits
  47. Two Simple Equations
  48. Universal Product Code
  49. Zeros and Nines

    Algebra

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

    Logic

  126. Lewis Carroll's Bilateral Diagrams
  127. Lewis Carroll's Trilateral Diagrams
  128. Lewis Carroll's Logic Game
  129. Venn Diagrams I
  130. Venn Diagrams II
  131. Formal Systems - an Example

    Fractals

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

    Probability

  151. Bertrand's Paradox
  152. Binomial Distribution
  153. Birds On a Wire
  154. Buffon's Noodle
  155. Chess Players Truel
  156. Conditional Probability and Independent Events
  157. Family Statistics
  158. Lewis Carroll's pillow problem
  159. Monty Hall Dilemma's Simulation
  160. Parrondo Paradox
  161. Random Clock Hands
  162. Stick Broken Into Three Pieces (Cartesian Coordinates)
  163. Stick Broken Into Three Pieces (Trilinear Coordinates)
  164. Who Needs Monty?

    Visual Illusions

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

    Calculus

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

    Math Magic

  233. An Arithmetic Magic Trick
  234. Arithmetic magic matrix
  235. Bachet's Magic Trick
  236. Barcode Magic
  237. Calendar Magic
  238. Gergonne's Magic Trick
  239. Hummer's Mind Reader
  240. Magic in Square
  241. Math Telepathy
  242. Number Guessing Game
  243. Two Numbers Guessing Game

    Combinatorics

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

    Combinatorial Games

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

    Puzzles & Games

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

    Geometric Fallacies

  371. A Circle With Two Centers
  372. All Triangles Are Isosceles
  373. Curry's Paradox
  374. Delian Problem Solved
  375. A Faulty Dissection
  376. Hooper's Paradox
  377. Is Every Trapezoid Parallelogram?
  378. John Sharp's Paradox
  379. Langman's Paradox
  380. Popping A Square
  381. Rouse Ball's Fallacy
  382. Sam Loyd's Son's Dissection
  383. Two Perpendiculars From a Point to a Line

    Social Science

  384. Adams' Apportionment Method
  385. Banzhaf Power Index Calculator
  386. Fair Division: Method of Lone Divider
  387. Fair Division: Method of Markers
  388. Fair Division: Method of Sealed Bids
  389. Fair Division: Method of Sealed Bids II
  390. Fast Power Indices
  391. Five Methods of Apportionment
  392. Four Voting Methods
  393. Hamilton's Apportionment Method
  394. Huntington-Hill Apportionment Method
  395. Jefferson's Apportionment Method
  396. Method of Markers II
  397. Shapley-Shubik Power Index Calculator
  398. Social Choice
  399. Webster's Apportionment Method
  400. Weighted Voting and Power Indices

    Elementary Geometry

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

    Stereometry

  521. Angles in a Cube I
  522. Angles in a Cube II
  523. Angles in a Cube III
  524. Bimedians in a Regular Tetrahedron
  525. Cut the Cone
  526. Cut the Cube
  527. Cut the Cylinder
  528. Regular Polyhedra - Platonic Solids
  529. Möbius' Strip
  530. Possible or Impossible?
  531. Stereographic Projection
  532. Three Pyramids are Better Than Two

    Outline (Fill-It-In) Mathematics

  533. 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. 60° Breeds 90°
  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 180°
  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 and 3-4-5 Triangles
  173. Equilateral Triangle, Straight Line and Tangent Circles
  174. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  175. Equilateral Triangles on Segments of Equilic Quadrilateral
  176. Equilateral Triangles On Sides of a Parallelogram
  177. Equilateral Triangles On Sides of a Parallelogram II
  178. Equilateral Triangles on Sides of a Quadrilateral
  179. Escher's Theorem
  180. Euler Line Cuts Off Equilateral Triangle
  181. Euclid I.43
  182. Euclid I.43 Extended
  183. Euclidean Construction of Center of Ellipse
  184. Euclidean Construction of Tangent to Ellipse
  185. Extouch Triangle in Poncelet Porism
  186. Eyeball Theorem
  187. External Angle Bisector
  188. Fagnano's problem
  189. Fagnano's problem II
  190. Fagnano's Problem, Third Solution
  191. Fagnano's Problem, Fourth Solution
  192. Fagnano's Problem in Reverse
  193. Fermat point and 9-point Centers
  194. Fermat's Point
  195. Fermat Points and Concurrent Euler Lines I
  196. Fermat Points and Concurrent Euler Lines II
  197. Feuerbach's theorem
  198. Feuerbach's Theorem: A Proof
  199. Find a Common Chord of Given Length
  200. First Vecten Point
  201. Five Cardioids
  202. Five Concyclic Points
  203. Five Incircles in a Square
  204. Five Incircles Theorem
  205. Fixed Point of Circles Orthogonal to the Given One
  206. Fixed point on a rod rotating around a triangle
  207. Focal Properties of Parabola
  208. Foot of Altitude and Minimum Distance
  209. Four Centroids and Parallels
  210. Four 9-Point Circles in a Quadrilateral
  211. Four Circles In a Triangle
  212. Four Concurrent Lines in a Cyclic Quadrilateral
  213. Four Incident Circles
  214. Four Incircles in Equilateral Triangle
  215. Four Pedal Circles
  216. Four Touching Circles
  217. Four Touching Circles II
  218. Four Touching Circles III
  219. Four Triangles, One Circle
  220. Four Turtles (or Bugs, or Dogs, etc.)
  221. Friendly Kiepert's Perspectors
  222. From Foci to a Tangent in Ellipse
  223. Fundamental Theorem of 3-Bar Motion
  224. Fuss' Theorem
  225. Generalizing a Romanian Olympiad Problem
  226. Geometric Optimization from the Asian Pacific Mathematical Olympiad
  227. Gergonne and Medial Triangles Are Orthologic
  228. Gergonne and Soddy Lines Are Perpendicular
  229. Gergonne in Ellipse
  230. Glide Reflection
  231. Golden Ratio in an Irregular Pentagon, Construction I
  232. Golden Ratio in an Irregular Pentagon, Construction II
  233. Golomb's inductive proof of a tromino theorem
  234. Gothic Arc
  235. Grebe: from Ladies' Diary to Carroll's Pillow
  236. Grebe's Theorem
  237. Greg Markowsky's Problem for Parabola
  238. Griffiths Points in Cyclic Quadrilateral
  239. Harmonic Ratio
  240. Hart Circle
  241. Hart's Inversor
  242. Haruki's Lemma
  243. Heron's Problem
  244. Hexagon Parallel to Medial Triangle
  245. Hexagon Parallel to Orthic Triangle
  246. Hinged Squares
  247. Hjelmslev Theorem
  248. Hofstadter Triangles and Points
  249. Homologous Lines under Three Spiral Similarities
  250. Homothety
  251. Hypocycloid Families
  252. Ian McGee's Observation
  253. Illustration of Pascal's Theorem
  254. Incenters in Cyclic Quadrilateral
  255. Incidence in Feuerbach's Theorem
  256. Incident Center Circles and Cardioid
  257. Incircles, Circumcircles and Inscribed Angles
  258. Inscriptible and Exscriptible Quadrilaterals
  259. Internal Tangents to Three Circles
  260. Intouch Triangle in Poncelet Porism
  261. Invariant Area Sweep Implies Pythagorean Theorem
  262. Inversion: Reflection in a Circle
  263. Inversion in the Incircle
  264. Inversion Tool
  265. Inversion with Negative Power
  266. Isogonal and Isotomic Conjugalities
  267. Isogonal Concurrency
  268. Isogonal image of the circumcircle
  269. Isosceles on the Sides of a Triangle
  270. Isotomy and Isogonality Hand-in-Hand
  271. Iterations in Geometry, an example
  272. Iterations in Geometry, Generalization
  273. Iterations on a Circle Through Three Points
  274. Iterations on Euler Lines
  275. Kiepert's Theorem
  276. Kürschak's Tile and Theorem
  277. Law of Cosines
  278. Law of Cosines: PWW
  279. Law of Cosines: An Unfolded Variant
  280. La Hire's Theorem
  281. La Hire's Theorem, a Variant
  282. La Hire's Theorem in Ellipse
  283. Lighthouse Curves
  284. Locus of Points in a Given Ratio to Two Points
  285. MacLaurin's Construction of Conics
  286. Malfatti's Problem
  287. Malfatti's Problem, Hart's Solution
  288. Maxwell's Theorem
  289. Median in Touching Circles
  290. Medians in a Quadrilateral
  291. Menelaus and Ceva Theorems
  292. Menelaus From Ceva
  293. Midline in Similar Triangles
  294. Midpoints from Gergonne Triangle
  295. Midpoints of the Lines Joining In- and Excenters
  296. Minkowski's addition of convex shapes
  297. Miquel's Point
  298. Miquel's Point of a 4-line Via Spiral Similarity
  299. Miquel's Theorem for Circles
  300. Mirror on the Wall
  301. Mirror property of the altitudes
  302. Mirror Property of Altitudes via Pascal's Hexagram
  303. Mixtilinear Circles and Concurrence
  304. Monge via Desargues
  305. Monge via Desargues II
  306. More Bottles in a Wine Rack
  307. Morley Constellation
  308. Morley's Theorem
  309. Morley's Theorem, Cardioids and Such
  310. Morley's Theorem, G. Zsolt Kiss' proof
  311. Nagel Line
  312. Nagel point
  313. Nagel Point of Medial Triangle
  314. Nagel's Theorem
  315. Napoleon-Barlotti Theorem
  316. Napoleon Theorem by Plane Tessellation
  317. Napoleon's Relatives
  318. Napoleon's Theorem by Transformation
  319. Neuberg Sangaku
  320. Newton's Construction of Conics
  321. Newton's Theorem
  322. Nine Point Circle
  323. No Equilateral Triangles, Please
  324. Nobbs' Points, Gergonne Line
  325. Non-classic Way To Draw Cardioid
  326. No-Pedal Collinearity
  327. On Bottema's Shoulders
  328. On Bottema's Shoulders II
  329. One Sided Surface in 4D
  330. Optical Property of Ellipse
  331. Optimization Problem in Acute Angle
  332. Orthocenter and Three Equal Circles
  333. Orthocentric System From Rectangle
  334. Orthogonality in Isosceles Triangles
  335. Orthologic Triangles in a Quadrilateral
  336. Orthopole
  337. Pairs of Homologous Lines under Spiral Similarities
  338. Pairs of Incircles in a Quadrilateral
  339. Pappus' Theorem
  340. Parabola As Envelope of Straight Lines
  341. Parabola As Envelope II
  342. Parabolic Mirror
  343. Paragon
  344. Parahexagon
  345. Parallel Chords
  346. Parallel Chords in Ellipse
  347. Parallel Lines and Similar Triangles
  348. Parallelogram and Similar Triangles
  349. Parallelogram Law
  350. Parallelogram Law: A PWW
  351. Parallelogram Law: A Tessellation
  352. Pascal in a Cyclic Quadrilateral
  353. Pascal Lines: Steiner and Kirkman Theorems
  354. Pascal Lines: Steiner and Kirkman Theorems II
  355. Pascal's Theorem
  356. Peaucellier Linkage
  357. Pedal Collinearities
  358. Pedal Polygons
  359. Pedal Triangle and Isogonal Conjugacy
  360. Pedoe's Theorem
  361. Perigal's Proof of the Pythagorean Proposition
  362. Perpendicular Bisectors in an Inscriptible Quadrilateral
  363. Perpendicular Bisectors in an Inscriptible Quadrilateral II
  364. Pinning Butterfly on Radical Axes
  365. Pivot Theorem
  366. Point common to two similar rectangles
  367. Polar Circle
  368. Pole and Polar with respect to a Triangle
  369. Poles and Polars
  370. Poles, Polars and Quadrilaterals
  371. Polygon from midpoints of its sides
  372. Polygons Formed by Perpendicular Bisectors
  373. Poncelet's Porism
  374. Poncelet's Porism in Ellipses
  375. Poncelet's Theorem
  376. Problem 4, 1975 USA Math Olympiad and the Radical Axis
  377. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
  378. Problem 4, 1975 USA Math Olympiad: Normals and Tangents
  379. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch
  380. Problem in an Equilateral Triangle
  381. Problem in an Equilateral Triangle II
  382. Problem In an Isosceles Right Triangle
  383. Problem in Three Squares
  384. Problem of Equal Steps
  385. Problem of Equal Steps II
  386. Problem of polygon construction
  387. Problem with Equilateral Triangles
  388. Product of Rotations
  389. Projections on Internal and External Angle Bisectors
  390. Projective Collinearity in a Quadrilateral
  391. Projective Generalization of Maxwell's Theorem
  392. Projective Proof of Pascal's Theorem
  393. Proof Without Words for the Pythagorean Theorem
  394. Properties of the Circle of Similitude
  395. Property of 6-Parpolygon
  396. Property of Angle Bisectors II
  397. Property of Circumscribed Quadrilaterals
  398. Property of Points Where In- and Excircles Touch a Triangle
  399. Property of the Line IO
  400. Property of the Line IO II
  401. Property of the Line IO: a proof from the Book

    Ptolemy on Hinges

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

    Miscellaneous Demonstrations

  567. Calculation of the Digits of pi by the Spigot Algorithm
  568. Calculation of the Digits of pi (Faster Version)
  569. Convex Polygon Is the Intersection of Half Planes
  570. Disappearing Lines
  571. Disappearing Rectangles
  572. Distance Between Strings
  573. Experimentation with Dynamic Geometry Software: An Example
  574. Genetic Algorithm Solves the Toads and Frogs Puzzle
  575. Impossible Fork
  576. Impossible Frame
  577. Morphing
  578. Projectile Trajectory
  579. Random Dot Stereograms
  580. Ratchet Effect
  581. Rotating Gears
  582. Sorting Algorithms
  583. Structural Constellation


Copyright © 1996-2010 Alexander Bogomolny

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