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

    Algebra

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

    Logic

  134. Lewis Carroll's Bilateral Diagrams
  135. Lewis Carroll's Trilateral Diagrams
  136. Lewis Carroll's Logic Game
  137. Venn Diagrams I
  138. Venn Diagrams II
  139. Formal Systems - an Example

    Fractals

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

    Probability

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

    Visual Illusions

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

    Calculus

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

    Math Magic

  241. An Arithmetic Magic Trick
  242. Arithmetic magic matrix
  243. Bachet's Magic Trick
  244. Barcode Magic
  245. Calendar Magic
  246. Gergonne's Magic Trick
  247. Hummer's Mind Reader
  248. Magic in Square
  249. Math Telepathy
  250. Number Guessing Game
  251. Two Numbers Guessing Game

    Combinatorics

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

    Combinatorial Games

  318. Date Game
  319. Dawson's Chess
  320. Dawson's Kayles
  321. Fraction Game
  322. Grundy's Game
  323. Hex 7
  324. Kayles
  325. Nim
  326. Nimble
  327. Northcott's game
  328. One Pile
  329. Plainim
  330. Plainim Misère
  331. Scoring
  332. Scoring Misère
  333. Scoring Misère: Two Heaps Perfect Strategy
  334. Silver Dollar Game
  335. Silver Dollar Game With No Silver Dollar
  336. Sticks, Jess realization of Scoring Misère
  337. Sticky Problem
  338. Sticky Problem II
  339. Subtraction Game
  340. TacTix
  341. Take-Away Games
  342. Turning Turtles
  343. Wythoff's Nim
  344. Wythoff's Nim II

    Puzzles & Games

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

    Geometric Fallacies

  385. A Circle With Two Centers
  386. All Triangles Are Isosceles
  387. Curry's Paradox
  388. Delian Problem Solved
  389. A Faulty Dissection
  390. Hooper's Paradox
  391. Is Every Trapezoid Parallelogram?
  392. John Sharp's Paradox
  393. Langman's Paradox
  394. Popping A Square
  395. Rouse Ball's Fallacy
  396. Sam Loyd's Son's Dissection
  397. Two Perpendiculars From a Point to a Line

    Social Science

  398. Adams' Apportionment Method
  399. Banzhaf Power Index Calculator
  400. Fair Division: Method of Lone Divider
  401. Fair Division: Method of Markers
  402. Fair Division: Method of Sealed Bids
  403. Fair Division: Method of Sealed Bids II
  404. Fast Power Indices
  405. Five Methods of Apportionment
  406. Four Voting Methods
  407. Hamilton's Apportionment Method
  408. Huntington-Hill Apportionment Method
  409. Jefferson's Apportionment Method
  410. Method of Markers II
  411. Shapley-Shubik Power Index Calculator
  412. Social Choice
  413. Webster's Apportionment Method
  414. Weighted Voting and Power Indices

    Elementary Geometry

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

    Miscellaneous Demonstrations

  544. Calculation of the Digits of pi by the Spigot Algorithm
  545. Calculation of the Digits of pi (Faster Version)
  546. Convex Polygon Is the Intersection of Half Planes
  547. Disappearing Lines
  548. Disappearing Rectangles
  549. Distance Between Strings
  550. Experimentation with Dynamic Geometry Software: An Example
  551. Genetic Algorithm Solves the Toads and Frogs Puzzle
  552. Impossible Fork
  553. Impossible Frame
  554. Morphing
  555. Projectile Trajectory
  556. Random Dot Stereograms
  557. Ratchet Effect
  558. Rotating Gears
  559. Sorting Algorithms
  560. Structural Constellation

 

    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 Circle Rolling in an Equilateral Triangle
  10. A Line in Triangle Through the Circumcenter
  11. A Parallelogram in Triangle
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku: Two Unrelated Circles
  14. A Sangaku with an Egyptian Attachment
  15. A Sangaku with Many Circles and Some
  16. A Stronger Triangle Inequality
  17. A Triangle of Antreas Hatzipolakis
  18. Adams' Circle
  19. Addition of Radius-Vectors
  20. Altitudes and the Power of a Point
  21. An Isoperimetric Theorem
  22. Angle Bisector in Equilateral Trapezoid
  23. Angle Bisector in Parallelogram
  24. Angle Bisector in Touching Circles
  25. Angle Bisectors and Perpendiculars in a Quadrilateral
  26. Angle Bisectors in a Quadrilateral
  27. Angle Preservation Property
  28. Angle Trisection by Archimedes
  29. Angle Trisection by Hippocrates
  30. Angle Trisection by Paul Vjecsner
  31. Angle Trisectors on Circumcircle
  32. Angles in Triangle Add to 180°
  33. Another Pair of Twins in an Arbelos
  34. Anticomplementary Triangle Surprise
  35. Antiparallel
  36. Apollonian Circles Theorem
  37. Apollonius Problem
  38. Application of Ceva's Theorem
  39. Archimedes' Quadruplets
  40. Archimedes Triangle and Squaring of Parabola
  41. Archimedes' Twin Circles and a Brother
  42. Area of an Arbelos
  43. Areas and Centroid in a Triangle
  44. Areas in Triangle
  45. Areas in Triangle II
  46. Arithmetic Mean Sangaku
  47. Arithmetic Mean Sangaku II
  48. Around the Incircle
  49. Around the Incircle in a Polygon
  50. Asian Pacific Kite
  51. Asymmetric Propeller I
  52. Asymmetric Propeller II
  53. Asymmetric Propeller III
  54. Barycentric Coordinates: a Tool
  55. Bevan's Point and Theorem
  56. Bisectal Circle
  57. Book of Lemmas: Proposition 1
  58. Book of Lemmas: Proposition 2
  59. Book of Lemmas: Proposition 3
  60. Book of Lemmas: Proposition 5
  61. Book of Lemmas: Proposition 6
  62. Book of Lemmas: Proposition 9
  63. Book of Lemmas: Proposition 10
  64. Book of Lemmas: Proposition 12
  65. Book of Lemmas: Proposition 13
  66. Böttcher's Proof of the Pythagorean Theorem
  67. Böttcher's Proof Variant
  68. Bottema Shatters Japan's Seclusion
  69. Bottema's Theorem
  70. Bottles in a Slanted Rack
  71. Bottles in a Wine Rack
  72. Bottles in a Wine Rack: Proofs and Generalizations
  73. Brahmagupta's Theorem
  74. Brianchon in Ellipse
  75. Brianchon's Theorem
  76. Bride's Chair
  77. Broken Chord Theorem
  78. Broken Chord Theorem by Paper Folding
  79. Broken Chord Theorem: G. Patruno's proof
  80. Broken Chord Theorem: S. Anderson's proof
  81. Butterfly in Ellipse
  82. Butterfly Problem
  83. Cabart's Collinearity
  84. Cantor's Theorem
  85. Cardioid As Envelope of Circles
  86. Cardioid As Epicycloid
  87. Carnot's Theorem
  88. Carpets in a Parallelogram
  89. Carpets in a Quadrilateral
  90. Carpets in a Quadrilateral II
  91. Carpets Theorem
  92. Carpets Theorem With Parallelograms
  93. Center Circles and Their Chains
  94. Centroids in a Polygon
  95. Ceva and Menelaus Theorems for Angle Bisectors
  96. Ceva's Theorem
  97. Ceva's Theorem (PWW)
  98. Cevian Nest
  99. Cevian Triangle
  100. Chain of Circles on a Chord
  101. Chain of Inscribed Circles
  102. Chasing Angles in Pascal's Hexagon
  103. Chasles' Theorem, a Proof
  104. Cherchez le quadrilatère cyclique
  105. Cherchez le quadrilatère cyclique II
  106. Chords, Concurrency and Orthic Triangle
  107. Chords in Cardioid
  108. Circle and Hyperbola as Lighthouse Curves
  109. Circle Chains and Inscribed Angles
  110. Circle Concurrency and Spiral Similarity
  111. Circle Crosses Triangle
  112. Circle Division by Euclid and Leonardo
  113. Circle Inscribed in a Circular Segment
  114. Circle of Similitude
  115. Circle through the Incenter
  116. Circle through the Incenter And Antiparallels
  117. Circle-Stacking Theorem
  118. Circle, Isosceles Triangle and Fixed Point
  119. Circles And Parallels
  120. Circles and Reflections in a Triangle
  121. Circles in Morley's Triangles
  122. Circles Lined on the Legs of a Right Triangle
  123. Circles On Cevians
  124. Circles Tangent to Circumcircle
  125. Circles through the Orthocenter
  126. Circles with Tangent Diameters
  127. Circumcevian Triangle
  128. Cleavance Center
  129. Cleaver
  130. Clifford Chain
  131. Clifford's Lemma
  132. Coaxal Circles on Perpendicular Bisector
  133. Coaxal Circles Theorem
  134. Colburn's Proof of the Pythagorean Theorem
  135. Common Tangents to Two Circles I
  136. Common Tangents to Two Circles II
  137. Complete Quadrilateral
  138. Concyclic Points in Equilateral Bumps
  139. Concurrency in Analytic Geometry
  140. Concurrency in Arbelos
  141. Concurrency in the Intouch Triangle
  142. Concyclic Circumcenters: A Dynamic View
  143. Concyclic Points in a Triangle
  144. Concyclic Points in Bride's Chair
  145. Conjugate Diameters in Ellipse
  146. Construction of Conics from Pascal's Theorem
  147. Consecutive Isosceles Decomposition
  148. Conway's Circle
  149. Copernicus' Theorem
  150. Crossed-Lines Construction of Shapes of Constant Width
  151. Cyclic Hexagon
  152. Cyclic Incenters
  153. Cycloids
  154. Diameters and Chords
  155. Directly Similar Figures
  156. Desargues' Theorem
  157. Discovery of Duane DeTemple
  158. Dividing a Segment into N parts: Besteman's Construction
  159. Dividing a Segment into N parts: Besteman's Construction II
  160. Dividing a Segment into N parts: GLaD Construction
  161. Dividing a Segment into N parts: SaRD Construction
  162. Dividing Evenly a Quadrilateral
  163. Dividing Evenly a Quadrilateral II
  164. Double Generation Theorem
  165. Douglas' Theorem (complex variables, linear algebra)
  166. Droz-Farny Circles
  167. Droz-Farny Line Theorem
  168. Dual to Pappus' Theorem
  169. Dynamic Construction of Ellipse and Other Curves
  170. Easy Construction of Bicentric Quadrilateral
  171. Easy Construction of Bicentric Quadrilateral II
  172. Eight Point Circle
  173. Ellipse, how to draw
  174. Ellipse in Arbelos
  175. Ellipse Between Two Circles
  176. Equal Angles in Two Circles
  177. Equal Angles in Two Circles II
  178. Equal Areas In Parallelogram à la Pythagoras
  179. Equal Incircles Theorem
  180. Equichordal Curves
  181. Equilateral and 3-4-5 Triangles
  182. Equilateral Triangle, Straight Line and Tangent Circles
  183. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  184. Equilateral Triangles on Segments of Equilic Quadrilateral
  185. Equilateral Triangles On Sides of a Parallelogram
  186. Equilateral Triangles On Sides of a Parallelogram II
  187. Equilateral Triangles on Sides of a Quadrilateral
  188. Escher's Theorem
  189. Euler Line Cuts Off Equilateral Triangle
  190. Euclid I.43
  191. Euclid I.43 Extended
  192. Euclidean Construction of Center of Ellipse
  193. Euclidean Construction of Tangent to Ellipse
  194. Extouch Triangle in Poncelet Porism
  195. Eyeball Theorem
  196. External Angle Bisector
  197. Fagnano's problem
  198. Fagnano's problem II
  199. Fagnano's Problem, Third Solution
  200. Fagnano's Problem, Fourth Solution
  201. Fagnano's Problem in Reverse
  202. Fermat point and 9-point Centers
  203. Fermat's Point
  204. Fermat Points and Concurrent Euler Lines I
  205. Fermat Points and Concurrent Euler Lines II
  206. Feuerbach's theorem
  207. Feuerbach's Theorem: A Proof
  208. Find a Common Chord of Given Length
  209. First Vecten Point
  210. Five Cardioids
  211. Five Concyclic Points
  212. Five Incircles in a Square
  213. Five Incircles Theorem
  214. Fixed Point of Circles Orthogonal to the Given One
  215. Fixed point on a rod rotating around a triangle
  216. Focal Properties of Parabola
  217. Foot of Altitude and Minimum Distance
  218. Four 9-Point Circles in a Quadrilateral
  219. Four Centroids and Parallels
  220. Four Circles In a Triangle
  221. Four Concurrent Lines in a Cyclic Quadrilateral
  222. Four Incident Circles
  223. Four Crossing Circles
  224. Four Incircles in Equilateral Triangle
  225. Four Pedal Circles
  226. Four Touching Circles
  227. Four Touching Circles II
  228. Four Touching Circles III
  229. Four Triangles, One Circle
  230. Four Turtles (or Bugs, or Dogs, etc.)
  231. Friendly Kiepert's Perspectors
  232. From Foci to a Tangent in Ellipse
  233. Fundamental Theorem of 3-Bar Motion
  234. Fuss' Theorem
  235. Generalizing a Romanian Olympiad Problem
  236. Geometric Optimization from the Asian Pacific Mathematical Olympiad
  237. Gergonne and Medial Triangles Are Orthologic
  238. Gergonne and Soddy Lines Are Perpendicular
  239. Gergonne in Ellipse
  240. Glide Reflection
  241. Golden Ratio in an Irregular Pentagon, Construction I
  242. Golden Ratio in an Irregular Pentagon, Construction II
  243. Golomb's inductive proof of a tromino theorem
  244. Gothic Arc
  245. Grebe: from Ladies' Diary to Carroll's Pillow
  246. Grebe's Theorem
  247. Greg Markowsky's Problem for Parabola
  248. Griffiths Points in Cyclic Quadrilateral
  249. Harmonic Ratio
  250. Hart Circle
  251. Hart's Inversor
  252. Haruki's Lemma
  253. Heron's Formula: a Proof
  254. Heron's Problem
  255. Hexagon Parallel to Medial Triangle
  256. Hexagon Parallel to Orthic Triangle
  257. Hinged Squares
  258. Hjelmslev Theorem
  259. Hofstadter Triangles and Points
  260. Homologous Lines under Three Spiral Similarities
  261. Homothety
  262. Hypocycloid Families
  263. Ian McGee's Observation
  264. Illustration of Pascal's Theorem
  265. Incenters in Cyclic Quadrilateral
  266. Incidence in Feuerbach's Theorem
  267. Incident Center Circles and Cardioid
  268. Incircles, Circumcircles and Inscribed Angles
  269. Inscriptible and Exscriptible Quadrilaterals
  270. Internal Tangents to Three Circles
  271. Intouch Triangle in Poncelet Porism
  272. Invariant Area Sweep Implies Pythagorean Theorem
  273. Inversion: Reflection in a Circle
  274. Inversion in the Incircle
  275. Inversion Tool
  276. Inversion with Negative Power
  277. Isogonal and Isotomic Conjugalities
  278. Isogonal Concurrency
  279. Isogonal image of the circumcircle
  280. Isosceles on the Sides of a Triangle
  281. Isotomy and Isogonality Hand-in-Hand
  282. Iterations in Geometry, an example
  283. Iterations in Geometry, Generalization
  284. Iterations on a Circle Through Three Points
  285. Iterations on Euler Lines
  286. Joseph Keech in Bride's Chair
  287. Kiepert's Theorem
  288. Kürschak's Tile and Theorem
  289. Law of Cosines
  290. Law of Cosines: PWW
  291. Law of Cosines: An Unfolded Variant
  292. La Hire's Theorem
  293. La Hire's Theorem, a Variant
  294. La Hire's Theorem in Ellipse
  295. Lighthouse Curves
  296. Line, Circle, and Fixed Points
  297. Locus of Points in a Given Ratio to Two Points
  298. MacLaurin's Construction of Conics
  299. Malfatti's Problem
  300. Malfatti's Problem, Hart's Solution
  301. Maximum Perimeter Property of the Incircle
  302. Maxwell's Theorem
  303. Median in Touching Circles
  304. Medians in a Quadrilateral
  305. Menelaus and Ceva Theorems
  306. Menelaus From Ceva
  307. Midline in Similar Triangles
  308. Midpoints from Gergonne Triangle
  309. Midpoints of the Lines Joining In- and Excenters
  310. Minkowski's addition of convex shapes
  311. Miquel's Point
  312. Miquel's Point of a 4-line Via Spiral Similarity
  313. Miquel's Theorem for Circles
  314. Mirror on the Wall
  315. Mirror property of the altitudes
  316. Mirror Property of Altitudes via Pascal's Hexagram
  317. Mixtilinear Circles and Concurrence
  318. Monge via Desargues
  319. Monge via Desargues II
  320. More Bottles in a Wine Rack
  321. Morley Constellation
  322. Morley's Theorem
  323. Morley's Theorem, Cardioids and Such
  324. Morley's Theorem, G. Zsolt Kiss' proof
  325. Nagel Line
  326. Nagel point
  327. Nagel Point of Medial Triangle
  328. Nagel's Theorem
  329. Napoleon-Barlotti Theorem
  330. Napoleon Theorem by Plane Tessellation
  331. Napoleon's Relatives
  332. Napoleon's Theorem by Transformation
  333. Neuberg Sangaku
  334. Newton's Construction of Conics
  335. Newton's Theorem
  336. Nine Point Circle
  337. No Equilateral Triangles, Please
  338. Nobbs' Points, Gergonne Line
  339. Non-classic Way To Draw Cardioid
  340. No-Pedal Collinearity
  341. On Bottema's Shoulders
  342. On Bottema's Shoulders II
  343. One Sided Surface in 4D
  344. Optical Property of Ellipse
  345. Optimization Problem in Acute Angle
  346. Orthocenter and Three Equal Circles
  347. Orthocentric System From Rectangle
  348. Orthogonality in Isosceles Triangles
  349. Orthologic Triangles in a Quadrilateral
  350. Orthopole
  351. Pairs of Homologous Lines under Spiral Similarities
  352. Pairs of Incircles in a Quadrilateral
  353. Pappus' Theorem
  354. Parabola As Envelope of Straight Lines
  355. Parabola As Envelope II
  356. Parabolic Mirror, Illustration
  357. Parabolic Mirror, Theory
  358. Paragon
  359. Parahexagon
  360. Parallel Chords
  361. Parallel Chords in Ellipse
  362. Parallel Lines and Similar Triangles
  363. Parallelogram and Similar Triangles
  364. Parallelogram Iterations
  365. Parallelogram Law
  366. Parallelogram Law: A PWW
  367. Parallelogram Law: A Tessellation
  368. Pascal in a Cyclic Quadrilateral
  369. Pascal Lines: Steiner and Kirkman Theorems
  370. Pascal Lines: Steiner and Kirkman Theorems II
  371. Pascal's Theorem
  372. Peaucellier Linkage
  373. Pedal Collinearities
  374. Pedal Polygons
  375. Pedal Triangle and Isogonal Conjugacy
  376. Pedoe's Theorem
  377. Perigal's Proof of the Pythagorean Proposition
  378. Perpendicular Bisectors in an Inscriptible Quadrilateral
  379. Perpendicular Bisectors in an Inscriptible Quadrilateral II
  380. Pinning Butterfly on Radical Axes
  381. Pivot Theorem
  382. Point common to two similar rectangles
  383. Polar Circle
  384. Pole and Polar with respect to a Triangle
  385. Poles and Polars
  386. Poles, Polars and Quadrilaterals
  387. Polygon from the midpoints of its sides
  388. Polygons Formed by Perpendicular Bisectors
  389. Poncelet's Porism
  390. Poncelet's Porism in Ellipses
  391. Poncelet's Theorem
  392. Problem 4, 1975 USA Math Olympiad and the Radical Axis
  393. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
  394. Problem 4, 1975 USA Math Olympiad: Normals and Tangents
  395. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch
  396. Problem in an Equilateral Triangle
  397. Problem in an Equilateral Triangle II
  398. Problem In an Isosceles Right Triangle
  399. Problem in Three Squares
  400. Problem of Equal Steps
  401. Problem of Equal Steps II
  402. Problem of polygon construction
  403. Problem with Equilateral Triangles
  404. Product of Rotations
  405. Projections on Internal and External Angle Bisectors
  406. Projective Collinearity in a Quadrilateral
  407. Projective Generalization of Maxwell's Theorem
  408. Projective Proof of Pascal's Theorem
  409. Proof Without Words for the Pythagorean Theorem
  410. Properties of the Circle of Similitude
  411. Property of 6-Parpolygon
  412. Property of Angle Bisectors II
  413. Property of Circumscribed Quadrilaterals
  414. Property of Points Where In- and Excircles Touch a Triangle
  415. Property of the Line IO
  416. Property of the Line IO II
  417. Property of the Line IO: a proof from the Book

    Ptolemy on Hinges

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

    Stereometry

  594. Angles in a Cube I
  595. Angles in a Cube II
  596. Angles in a Cube III
  597. Bimedians in a Regular Tetrahedron
  598. Cut the Cone
  599. Cut the Cube
  600. Cut the Cylinder
  601. Regular Polyhedra - Platonic Solids
  602. Other Solids
  603. Möbius' Strip
  604. Possible or Impossible?
  605. Stereographic Projection
  606. Three Pyramids are Better Than Two

    Outline (Fill-It-In) Mathematics

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


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