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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. 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. Listing All the Composite Numbers
  37. Lucky Numbers
  38. Make An Identity
  39. Make It All Zeros
  40. Napier Bones
  41. Plain Multiplication table up to 20×20
  42. Product of Simple Fractions
  43. Sieve of Eratosthenes
  44. Sieve of Squares
  45. Soroban in Various Number Systems
  46. Suan Pan in Various Number Systems
  47. Sums and Products
  48. Sums of Powers of Digits
  49. Two Simple Equations
  50. Universal Product Code
  51. Zeros and Nines

    Algebra

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

    Logic

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

    Fractals

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

    Probability

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

    Visual Illusions

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

    Calculus

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

    Math Magic

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

    Combinatorics

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

    Combinatorial Games

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

    Puzzles & Games

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

    Geometric Fallacies

  378. A Circle With Two Centers
  379. All Triangles Are Isosceles
  380. Curry's Paradox
  381. Delian Problem Solved
  382. A Faulty Dissection
  383. Hooper's Paradox
  384. Is Every Trapezoid Parallelogram?
  385. John Sharp's Paradox
  386. Langman's Paradox
  387. Popping A Square
  388. Rouse Ball's Fallacy
  389. Sam Loyd's Son's Dissection
  390. Two Perpendiculars From a Point to a Line

    Social Science

  391. Adams' Apportionment Method
  392. Banzhaf Power Index Calculator
  393. Fair Division: Method of Lone Divider
  394. Fair Division: Method of Markers
  395. Fair Division: Method of Sealed Bids
  396. Fair Division: Method of Sealed Bids II
  397. Fast Power Indices
  398. Five Methods of Apportionment
  399. Four Voting Methods
  400. Hamilton's Apportionment Method
  401. Huntington-Hill Apportionment Method
  402. Jefferson's Apportionment Method
  403. Method of Markers II
  404. Shapley-Shubik Power Index Calculator
  405. Social Choice
  406. Webster's Apportionment Method
  407. Weighted Voting and Power Indices

    Elementary Geometry

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

    Stereometry

  528. Angles in a Cube I
  529. Angles in a Cube II
  530. Angles in a Cube III
  531. Bimedians in a Regular Tetrahedron
  532. Cut the Cone
  533. Cut the Cube
  534. Cut the Cylinder
  535. Regular Polyhedra - Platonic Solids
  536. Möbius' Strip
  537. Possible or Impossible?
  538. Stereographic Projection
  539. Three Pyramids are Better Than Two

 

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

    Ptolemy on Hinges

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

    Miscellaneous Demonstrations

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

    Outline (Fill-It-In) Mathematics

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


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