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Manifesto: what CTK is about Buying a book is a commitment to learning Table of content Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page
Memory and MatchingArithmeticAlgebra
LogicFractalsProbability
Visual IllusionsCalculusMath Magic
CombinatoricsCombinatorial GamesPuzzles & Games
Geometric FallaciesSocial SciencesElementary Geometry
StereometryOutline MathematicsMiscellaneous Demonstrations
Interactive Mathematics Activities

    Memory and Matching

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

    Arithmetic

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

    Algebra

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

    Logic

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

    Fractals

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

    Probability

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

    Visual Illusions

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

    Calculus

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

    Math Magic

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

    Combinatorics

  234. 3-Colors Tower of Hanoi
  235. Affirmative Action Problem
  236. Balanced Incomplete Block Design on a Triangular Grid
  237. Bicolor Tower of Hanoi
  238. Blithe 12
  239. Cheapest Link & Kruskal's Algorithms - skill practice
  240. Chvatal's Art Gallery Theorem
  241. Coin change
  242. Coloring a Graph
  243. Critical Path in a Weighted Digraph
  244. de Bruijn Cycle
  245. Dominoes on a Chessboard
  246. Euler & Hamilton Paths - skill practice
  247. Five Lattice Points
  248. Flipping Items Simultaneously II
  249. Flipping pancakes
  250. Gray Codes
  251. Group multiplication of permutations
  252. Hanoing
  253. Happy 8
  254. Hex Can't End in a Draw
  255. How to define a permutation
  256. Infinite Latin Square
  257. Jumping Isn't Everything
  258. Latin Squares
  259. Leibniz and Pascal Triangles
  260. Life-like Automaton With Definable Rules
  261. Light Bulbs in a Circle
  262. Listing All Combinations
  263. Listing All Permutations
  264. Listing All Permutations: Johnson-Trotter Method
  265. Lucky 7
  266. Make Your Move, Kid!
  267. Marriage Problem, simulation
  268. Mazes
  269. Monotone Number Sequences
  270. Number sequences and Pigeonhole
  271. Party Acquaintances
  272. Permutations as a Product of Transpositions
  273. Product of Permutations
  274. Project Scheduling
  275. Property of the Powers of 2
  276. Recombining Permutations
  277. Sam Loyd's Fifteen
  278. Scrub Tile Puzzle
  279. Shuttle Puzzle
  280. Simple Graphs
  281. Sliders (slider games on torus, Moebius strip, projective plane, permutations, cycles)
  282. Straight Tromino on a Chessboard
  283. The Game of Life
  284. Tool for Constructing Orthogonal Latin Squares
  285. Tower of Hanoi
  286. Tower of Hanoi, the Hard Way
  287. Two Men of Tibet
  288. Universal Coloring
  289. Y Can't End in a Draw

    Combinatorial Games

  290. Date Game
  291. Dawson's Chess
  292. Dawson's Kayles
  293. Fraction Game
  294. Grundy's Game
  295. Hex 7
  296. Kayles
  297. Nim
  298. Nimble
  299. Northcott's game
  300. One Pile
  301. Plainim
  302. Plainim Misère
  303. Scoring
  304. Scoring Misère
  305. Silver Dollar Game
  306. Silver Dollar Game With No Silver Dollar
  307. Sticks, Jess realization of Scoring Misère
  308. Sticky Problem
  309. Sticky Problem II
  310. Subtraction Game
  311. TacTix
  312. Take-Away Games
  313. Turning Turtles
  314. Wythoff's Nim
  315. Wythoff's Nim II

    Puzzles & Games

  316. 3-Chain
  317. 4×4×4 Tic Tac Toe (spacial visualization)
  318. Achieve the Equilibrium
  319. Bulls and Cows
  320. Caterpillar
  321. Changing Colors (combinatorics, invariance)
  322. Color Balls
  323. Computer Mastermind (logic and problem solving)
  324. Eight Digits Puzzle
  325. Farmer and Wife To Catch Rooster and Hen (invariance, city block distance)
  326. Fif (multiple representations)
  327. Fill a Hexagonal Grid (space visualization)
  328. Fish Soup Game (multiple representations)
  329. Four Knights (multiple representations)
  330. Full House
  331. Goat, Cabbage and Wolf to move across a river (multiple representations)
  332. Hare and Hounds
  333. Jam (multiple representations)
  334. Looping Chips (multiple representations)
  335. Mastermind (logic and problem solving)
  336. Mastermind Variants (logic and problem solving)
  337. Mirror Maze
  338. Peg Solitaire (logic, problem solving, subproblems)
  339. Pick A Label (logic, problem solving)
  340. Reverse Solitaire (logic, problem solving, subproblems)
  341. Sending scouts into the desert (algebra, geometric series, problem solving)
  342. Seven Coins Puzzle (multiple representations)
  343. Shuttle Puzzle Practice
  344. Solitaire on a Circle (iterations, invariance)
  345. Stones in Cups
  346. Super Nim
  347. Thugs and Bowmen
  348. Tiling a Triangulated Hexagon (a proof without words)
  349. Toads And Frogs in 2 dimensions (repeated application of same idea)
  350. Tromino Puzzle
  351. Vector Addition and Subtraction
  352. Vowel Movement
  353. Wolf and Rabbit

    Geometric Fallacies

  354. A Circle With Two Centers
  355. All Triangles Are Isosceles
  356. Curry's Paradox
  357. Delian Problem Solved
  358. A Faulty Dissection
  359. Hooper's Paradox
  360. Is Every Trapezoid Parallelogram?
  361. John Sharp's Paradox
  362. Langman's Paradox
  363. Popping A Square
  364. Rouse Ball's Fallacy
  365. Sam Loyd's Son's Dissection
  366. Two Perpendiculars From a Point to a Line

    Social Science

  367. Adams' Apportionment Method
  368. Fair Division: Method of Lone Divider
  369. Fair Division: Method of Markers
  370. Fair Division: Method of Sealed Bids
  371. Fast Power Indices
  372. Hamilton's Apportionment Method
  373. Huntington-Hill Apportionment Method
  374. Jefferson's Apportionment Method
  375. Social Choice
  376. Webster's Apportionment Method
  377. Weighted Voting and Power Indices

    Elementary Geometry

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

    Stereometry

  483. Angles in a Cube I
  484. Angles in a Cube II
  485. Angles in a Cube III
  486. Bimedians in a Regular Tetrahedron
  487. Cut the Cone
  488. Cut the Cube
  489. Cut the Cylinder
  490. Regular Polyhedra - Platonic Solids
  491. Möbius' Strip
  492. Stereographic Projection
  493. Three Pyramids are Better Than Two

    Outline (Fill-It-In) Mathematics

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

 

    Geometry

  1. 120° Breeds 90°
  2. 2N-Wing Butterfly Theorem
  3. 3 Isosceles Trapezoids
  4. 5-Star and A Circle
  5. 6 to 9 Point Circle
  6. 60o Breeds 90o
  7. 9-point Circle as a Locus of Concurrency
  8. A Circle Related to Incenter and Circumcenter
  9. A Line in Triangle Through the Circumcenter
  10. A Parallelogram in Triangle
  11. A Sangaku Follow-Up on an Archimedes' Lemma
  12. A Sangaku: Two Unrelated Circles
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Stronger Triangle Inequality
  16. A Triangle of Antreas Hatzipolakis
  17. Adams' Circle
  18. Addition of Radius-Vectors
  19. Altitudes and the Power of a Point
  20. An Isoperimetric Theorem
  21. Angle Bisector in Parallelogram
  22. Angle Bisector in Touching Circles
  23. Angle Bisectors and Perpendiculars in a Quadrilateral
  24. Angle Bisectors in a Quadrilateral
  25. Angle Preservation Property
  26. Angle Trisection by Archimedes
  27. Angle Trisection by Hippocrates
  28. Angle Trisection by Paul Vjecsner
  29. Angle Trisectors on Circumcircle
  30. Angles in Triangle Add to 180o
  31. Another Pair of Twins in an Arbelos
  32. Anticomplementary Triangle Surprise
  33. Antiparallel
  34. Apollonian Circles Theorem
  35. Apollonius Problem
  36. Application of Ceva's Theorem
  37. Archimedes' Quadruplets
  38. Archimedes Triangle and Squaring of Parabola
  39. Archimedes' Twin Circles and a Brother
  40. Area of an Arbelos
  41. Areas and Centroid in a Triangle
  42. Areas in Triangle
  43. Areas in Triangle II
  44. Arithmetic Mean Sangaku
  45. Arithmetic Mean Sangaku II
  46. Around the Incircle
  47. Around the Incircle in a Polygon
  48. Asymmetric Propeller I
  49. Asymmetric Propeller II
  50. Asymmetric Propeller III
  51. Barycentric Coordinates: a Tool
  52. Bevan's Point and Theorem
  53. Bisectal Circle
  54. Book of Lemmas: Proposition 1
  55. Book of Lemmas: Proposition 2
  56. Book of Lemmas: Proposition 3
  57. Book of Lemmas: Proposition 5
  58. Book of Lemmas: Proposition 6
  59. Book of Lemmas: Proposition 9
  60. Book of Lemmas: Proposition 10
  61. Book of Lemmas: Proposition 12
  62. Book of Lemmas: Proposition 13
  63. Böttcher's Proof of the Pythagorean Theorem
  64. Böttcher's Proof Variant
  65. Bottema Shatters Japan's Seclusion
  66. Bottema's Theorem
  67. Bottles in a Slanted Rack
  68. Bottles in a Wine Rack
  69. Bottles in a Wine Rack: Proofs and Generalizations
  70. Brahmagupta's Theorem
  71. Brianchon in Ellipse
  72. Brianchon's Theorem
  73. Bride's Chair
  74. Broken Chord Theorem
  75. Broken Chord Theorem by Paper Folding
  76. Broken Chord Theorem: G. Patruno's proof
  77. Broken Chord Theorem: S. Anderson's proof
  78. Butterfly in Ellipse
  79. Butterfly Problem
  80. Cantor's Theorem
  81. Cardioid As Envelope of Circles
  82. Cardioid As Epicycloid
  83. Carnot's Theorem
  84. Carpets in a Parallelogram
  85. Carpets in a Quadrilateral
  86. Carpets in a Quadrilateral II
  87. Carpets Theorem
  88. Carpets Theorem With Parallelograms
  89. Center Circles and Their Chains
  90. Centroids in a Polygon
  91. Ceva and Menelaus Theorems for Angle Bisectors
  92. Ceva's Theorem
  93. Ceva's Theorem (PWW)
  94. Cevian Nest
  95. Cevian Triangle
  96. Chain of Inscribed Circles
  97. Chasles' Theorem, a Proof
  98. Cherchez le quadrilatère cyclique
  99. Cherchez le quadrilatère cyclique II
  100. Chords, Concurrency and Orthic Triangle
  101. Chords in Cardioid
  102. Circle and Hyperbola as Lighthouse Curves
  103. Circle Chains and Inscribed Angles
  104. Circle Concurrency and Spiral Similarity
  105. Circle Crosses Triangle
  106. Circle Division by Euclid and Leonardo
  107. Circle Inscribed in a Circular Segment
  108. Circle of Similitude
  109. Circle-Stacking Theorem
  110. Circles And Parallels
  111. Circles and Reflections in a Triangle
  112. Circles in Morley's Triangles
  113. Circles Lined on the Legs of a Right Triangle
  114. Circles On Cevians
  115. Circles Tangent to Circumcircle
  116. Circles through the Orthocenter
  117. Circles with Tangent Diameters
  118. Circumcevian Triangle
  119. Cleavance Center
  120. Cleaver
  121. Clifford Chain
  122. Clifford's Lemma
  123. Coaxal Circles Theorem
  124. Colburn's Proof of the Pythagorean Theorem
  125. Common Tangents to Two Circles I
  126. Common Tangents to Two Circles II
  127. Complete Quadrilateral
  128. Concyclic Points in Equilateral Bumps
  129. Concurrency in Analytic Geometry
  130. Concurrency in Arbelos
  131. Concurrency in the Intouch Triangle
  132. Concyclic Circumcenters: A Dynamic View
  133. Concyclic Points in a Triangle
  134. Conjugate Diameters in Ellipse
  135. Construction of Conics from Pascal's Theorem
  136. Consecutive Isosceles Decomposition
  137. Conway's Circle
  138. Copernicus' Theorem
  139. Crossed-Lines Construction of Shapes of Constant Width
  140. Cyclic Hexagon
  141. Cyclic Incenters
  142. Cycloids
  143. Diameters and Chords
  144. Directly Similar Figures
  145. Desargues' Theorem
  146. Discovery of Duane DeTemple
  147. Dividing a Segment into N parts: Besteman's Construction
  148. Dividing a Segment into N parts: Besteman's Construction II
  149. Dividing a Segment into N parts: GLaD Construction
  150. Dividing a Segment into N parts: SaRD Construction
  151. Dividing Evenly a Quadrilateral
  152. Dividing Evenly a Quadrilateral II
  153. Double Generation Theorem
  154. Douglas' Theorem (complex variables, linear algebra)
  155. Droz-Farny Circles
  156. Droz-Farny Line Theorem
  157. Dual to Pappus' Theorem
  158. Dynamic Construction of Ellipse and Other Curves
  159. Easy Construction of Bicentric Quadrilateral
  160. Easy Construction of Bicentric Quadrilateral II
  161. Eight Point Circle
  162. Ellipse, how to draw
  163. Ellipse in Arbelos
  164. Ellipse Between Two Circles
  165. Equal Angles in Two Circles
  166. Equal Angles in Two Circles II
  167. Equal Areas In Parallelogram à la Pythagoras
  168. Equal Incircles Theorem
  169. Equichordal Curves
  170. Equilateral Triangle, Straight Line and Tangent Circles
  171. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  172. Equilateral Triangles on Segments of Equilic Quadrilateral
  173. Equilateral Triangles On Sides of a Parallelogram
  174. Equilateral Triangles On Sides of a Parallelogram II
  175. Equilateral Triangles on Sides of a Quadrilateral
  176. Euler Line Cuts Off Equilateral Triangle
  177. Euclid I.43
  178. Euclid I.43 Extended
  179. Euclidean Construction of Center of Ellipse
  180. Euclidean Construction of Tangent to Ellipse
  181. Extouch Triangle in Poncelet Porism
  182. Eyeball Theorem
  183. External Angle Bisector
  184. Fagnano's problem
  185. Fagnano's problem II
  186. Fagnano's Problem, Third Solution
  187. Fagnano's Problem, Fourth Solution
  188. Fagnano's Problem in Reverse
  189. Fermat point and 9-point Centers
  190. Fermat's Point
  191. Feuerbach's theorem
  192. Feuerbach's Theorem: A Proof
  193. First Vecten Point
  194. Five Cardioids
  195. Five Concyclic Points
  196. Five Incircles in a Square
  197. Five Incircles Theorem
  198. Fixed Point of Circles Orthogonal to the Given One
  199. Fixed point on a rod rotating around a triangle
  200. Focal Properties of Parabola
  201. Foot of Altitude and Minimum Distance
  202. Four Centroids and Parallels
  203. Four 9-Point Circles in a Quadrilateral
  204. Four Circles In a Triangle
  205. Four Concurrent Lines in a Cyclic Quadrilateral
  206. Four Incident Circles
  207. Four Incircles in Equilateral Triangle
  208. Four Pedal Circles
  209. Four Touching Circles
  210. Four Touching Circles II
  211. Four Touching Circles III
  212. Four Triangles, One Circle
  213. Four Turtles (or Bugs, or Dogs, etc.)
  214. Friendly Kiepert's Perspectors
  215. From Foci to a Tangent in Ellipse
  216. Fundamental Theorem of 3-Bar Motion
  217. Fuss' Theorem
  218. Generalizing a Romanian Olympiad Problem
  219. Gergonne and Medial Triangles Are Orthologic
  220. Gergonne and Soddy Lines Are Perpendicular
  221. Gergonne in Ellipse
  222. Glide Reflection
  223. Golden Ratio in an Irregular Pentagon, Construction I
  224. Golden Ratio in an Irregular Pentagon, Construction II
  225. Golomb's inductive proof of a tromino theorem
  226. Gothic Arc
  227. Grebe: from Ladies' Diary to Carroll's Pillow
  228. Grebe's Theorem
  229. Greg Markowsky's Problem for Parabola
  230. Griffiths Points in Cyclic Quadrilateral
  231. Harmonic Ratio
  232. Hart Circle
  233. Hart's Inversor
  234. Haruki's Lemma
  235. Heron's Problem
  236. Hexagon Parallel to Medial Triangle
  237. Hexagon Parallel to Orthic Triangle
  238. Hinged Squares
  239. Hjelmslev Theorem
  240. Hofstadter Triangles and Points
  241. Homologous Lines under Three Spiral Similarities
  242. Homothety
  243. Hypocycloid Families
  244. Ian McGee's Observation
  245. Illustration of Pascal's Theorem
  246. Incenters in Cyclic Quadrilateral
  247. Incidence in Feuerbach's Theorem
  248. Incident Center Circles and Cardioid
  249. Incircles, Circumcircles and Inscribed Angles
  250. Inscriptible and Exscriptible Quadrilaterals
  251. Internal Tangents to Three Circles
  252. Intouch Triangle in Poncelet Porism
  253. Inversion: Reflection in a Circle
  254. Inversion in the Incircle
  255. Inversion Tool
  256. Inversion with Negative Power
  257. Isogonal and Isotomic Conjugalities
  258. Isogonal Concurrency
  259. Isogonal image of the circumcircle
  260. Isosceles on the Sides of a Triangle
  261. Isotomy and Isogonality Hand-in-Hand
  262. Iterations in Geometry, an example
  263. Iterations in Geometry, Generalization
  264. Iterations on a Circle Through Three Points
  265. Iterations on Euler Lines
  266. Kiepert's Theorem
  267. Kürschak's Tile and Theorem
  268. Law of Cosines
  269. Law of Cosines: PWW
  270. Law of Cosines: An Unfolded Variant
  271. La Hire's Theorem
  272. La Hire's Theorem, a Variant
  273. La Hire's Theorem in Ellipse
  274. Lean Napoleon's Triangles
  275. Lighthouse Curves
  276. Locus of Points in a Given Ratio to Two Points
  277. MacLaurin's Construction of Conics
  278. Malfatti's Problem
  279. Malfatti's Problem, Hart's Solution
  280. Maxwell's Theorem
  281. Median in Touching Circles
  282. Medians in a Quadrilateral
  283. Menelaus and Ceva Theorems
  284. Menelaus From Ceva
  285. Midline in Similar Triangles
  286. Midpoints from Gergonne Triangle
  287. Midpoints of the Lines Joining In- and Excenters
  288. Minkowski's addition of convex shapes
  289. Miquel's Point
  290. Miquel's Point of a 4-line Via Spiral Similarity
  291. Miquel's Theorem for Circles
  292. Mirror on the Wall
  293. Mirror property of the altitudes
  294. Mirror Property of Altitudes via Pascal's Hexagram
  295. Mixtilinear Circles and Concurrence
  296. Monge via Desargues
  297. Monge via Desargues II
  298. More Bottles in a Wine Rack
  299. Morley Constellation
  300. Morley's Theorem
  301. Morley's Theorem, Cardioids and Such
  302. Morley's Theorem, G. Zsolt Kiss' proof
  303. Nagel Line
  304. Nagel point
  305. Nagel Point of Medial Triangle
  306. Nagel's Theorem
  307. Napoleon Theorem by Plane Tessellation
  308. Napoleon's Theorem by Transformation
  309. Neuberg Sangaku
  310. Newton's Construction of Conics
  311. Newton's Theorem
  312. Nine Point Circle
  313. No Equilateral Triangles, Please
  314. Nobbs' Points, Gergonne Line
  315. Non-classic Way To Draw Cardioid
  316. No-Pedal Collinearity
  317. On Bottema's Shoulders
  318. On Bottema's Shoulders II
  319. One Sided Surface in 4D
  320. Optical Property of Ellipse
  321. Optimization Problem in Acute Angle
  322. Orthocenter and Three Equal Circles
  323. Orthocentric System From Rectangle
  324. Orthogonality in Isosceles Triangles
  325. Orthologic Triangles in a Quadrilateral
  326. Orthopole
  327. Pairs of Homologous Lines under Spiral Similarities
  328. Pairs of Incircles in a Quadrilateral
  329. Pappus' Theorem
  330. Parabola As Envelope of Straight Lines
  331. Parabola As Envelope II
  332. Parabolic Mirror
  333. Paragon
  334. Parahexagon
  335. Parallel Chords
  336. Parallel Chords in Ellipse
  337. Parallel Lines and Similar Triangles
  338. Parallelogram and Similar Triangles
  339. Parallelogram Law
  340. Parallelogram Law: A PWW
  341. Parallelogram Law: A Tessellation
  342. Pascal in a Cyclic Quadrilateral
  343. Pascal Lines: Steiner and Kirkman Theorems
  344. Pascal Lines: Steiner and Kirkman Theorems II
  345. Pascal's Theorem
  346. Peaucellier Linkage
  347. Pedal Collinearities
  348. Pedal Polygons
  349. Pedal Triangle and Isogonal Conjugacy
  350. Pedoe's Theorem
  351. Perigal's Proof of the Pythagorean Proposition
  352. Perpendicular Bisectors in an Inscriptible Quadrilateral
  353. Perpendicular Bisectors in an Inscriptible Quadrilateral II
  354. Pinning Butterfly on Radical Axes
  355. Pivot Theorem
  356. Point common to two similar rectangles
  357. Polar Circle
  358. Pole and Polar with respect to a Triangle
  359. Poles and Polars
  360. Poles, Polars and Quadrilaterals
  361. Polygon from midpoints of its sides
  362. Polygons Formed by Perpendicular Bisectors
  363. Poncelet's Porism
  364. Poncelet's Porism in Ellipses
  365. Poncelet's Theorem
  366. Problem in an Equilateral Triangle
  367. Problem in an Equilateral Triangle II
  368. Problem In an Isosceles Right Triangle
  369. Problem in Three Squares
  370. Problem of Equal Steps
  371. Problem of Equal Steps II
  372. Problem of polygon construction
  373. Problem with Equilateral Triangles
  374. Product of Rotations
  375. Projections on Internal and External Angle Bisectors
  376. Projective Collinearity in a Quadrilateral
  377. Projective Generalization of Maxwell's Theorem
  378. Projective Proof of Pascal's Theorem
  379. Proof Without Words for the Pythagorean Theorem
  380. Properties of the Circle of Similitude
  381. Property of 6-Parpolygon
  382. Property of Angle Bisectors II
  383. Property of Circumscribed Quadrilaterals
  384. Property of Points Where In- and Excircles Touch a Triangle
  385. Property of the Line IO
  386. Property of the Line IO II
  387. Property of the Line IO: a proof from the Book
  388. Pythagoras and Vecten Break Japan's Isolation
  389. Pythagorean Condition in An Isosceles Right Triangle
  390. Pythagorean Theorem: Subtle Dangers of Visual Proof
  391. Pythagorean Theorem By Euclid
  392. Pythagorean Theorem By Plane Tessellation
  393. Pythagorean Theorem By Hexagonal Tessellation
  394. Pythagorean Theorem By Rearrangement
  395. Pythagorean Theorem By Shearing and Rotations I
  396. Pythagorean Theorem By Shearing and Rotations II
  397. Pythagorean Theorem in an Orthodiagonal Quadrilateral
  398. Quadrilaterals Formed by Perpendicular Bisectors
  399. Quest for Paragon
  400. MathPro Logo
  401. Radical Axis and Center, an Application
  402. Radical Axis of Circles Inscribed in a Circular Segment
  403. Radical Center
  404. Radius and Construction of a Mixtilinear Circle
  405. Radius of a Circle by Paper Folding
  406. Rectangle in Arbelos
  407. Reflection In Line
  408. Reflections in Ellipse
  409. Reflections of a Line Through the Orthocenter
  410. Reflections of a Point on the Circumcircle
  411. Reflections of the Orthocenter
  412. Remarkable Line in a Quadrilateral
  413. Remarkable Line in Cyclic Quadrilateral
  414. Rendezvous
  415. Right Isosceles Triangles on Sides of a Quadrilateral
  416. Rotation Transform
  417. Rotations in Disguise
  418. Rusty Compass Construction of Equilateral Triangle
  419. Salinon: from Archimedes Book of Lemmas
  420. Sangaku Iterations, Is it Wasan?
  421. Sangakus with a Mixtilinear Circle
  422. Sangaku with Three Mixtilinear Circles
  423. Seven Circles Theorem
  424. Squares in Arbelos
  425. Secant, Tangents and Orthogonality
  426. See-Saw Lemma
  427. Segment Trisection
  428. Shape of Constant Width
  429. Shearing Butterflies in Quadrilaterals
  430. Siamese Triangles
  431. Siamese Triangles II
  432. Similar Triangles on Sides of a Quadrilateral
  433. Similar Triangles on Sides and Diagonals of a Quadrilateral
  434. Simson Line
  435. Simson Line From Isogonal Perspective
  436. Simson Lines Of Diametrically Opposite Points
  437. Simsons and 9-Point Circles in Cyclic Quadrilateral
  438. sin(N)
  439. Six Circles Theorem (Elkies)
  440. Six Circles Theorem (Evelyn)
  441. Six Concyclic Points
  442. Six Concyclic Points II
  443. Six Incircles in an Equilateral Triangle
  444. Six Point Circle
  445. SixPoints, ThreeLines
  446. Soddy Circles and David Eppstein's Centers
  447. Sperner's Lemma
  448. Spira Mirabilis - Wonderful Spiral
  449. Spiral Similarity
  450. Square in a Right Triangle
  451. Square Inscribed in a Triangle
  452. Square Inscribed in a Triangle II
  453. Square Inscribed In a Triangle III
  454. Squares in Circles
  455. Squares on Sides of a Quadrilateral
  456. Squinting Eyes Theorem
  457. Star Construction of Shapes of Constant Width
  458. Steiner's Chain
  459. Steiner's porism
  460. Straight Edge Construction of Polar
  461. Symmedian and Antiparallel
  462. Symmedian and 2 Antiparallels
  463. Symmedian and the Tangents
  464. Symmedian in a Right Triangle
  465. Tangency Point of Two Circles
  466. Tangent and Secant
  467. Tangent Circles and an Isosceles Triangle Tangent Circles and an Isosceles Triangle II
  468. Tangent Circles in a Parallelogram
  469. Tangent in Concurrency
  470. Tangents and Diagonals in Cyclic Quadrilateral
  471. Tangents, Perpendiculars and Geometric Mean
  472. Tesseract
  473. Thébault's Problem I
  474. Thébault's Problem II
  475. Thébault's Problem III
  476. Thébault's Problem IV
  477. Thébault's Problem III, Proof
  478. Theorem of Complete Quadrilateral
  479. Three Concurrent Circles
  480. Three Congruent Circles by Reflection
  481. Three Congruent Circles by Reflection II
  482. Three Congruent Circles by Reflection III
  483. Three Incircles In a Right Triangle
  484. Three Incircles in a Triangle
  485. Three Isosceles Triangles
  486. Three Parabola Tangents
  487. Three Parallels in a Triangle
  488. Three Points on a Parabola
  489. Three Roads, Three Travelers
  490. Three Similar Polygons
  491. Three Similar Triangles
  492. Three Similar Triangles (but differently oriented)
  493. Three Similar Triangles (but differently oriented) II
  494. Three Squares and Two Ellipses
  495. Three Tangent Circles
  496. Three Tangents Theorem
  497. Three Tangents Theorem II
  498. Three Tangents, Three Chords in Ellipse
  499. Three Tangents, Three Secants
  500. Toothpick Construction of a Square
  501. Touching Circles
  502. Touching Circles and Concurrency
  503. Triangle and Segment Count in a Polygon
  504. Triangle Geometry
  505. Triangle Sliding on Two Straight Lines
  506. Triangles, Squares and Areas from Temple Geometry
  507. Triangles with Equal Area
  508. Triangular Billiard
  509. Trigonometric Form of Ceva's Theorem (a tool for discovery of trigonometric identities and other problems)
  510. Tromino as a Rep-tile
  511. Tucker Circles
  512. Tucker Circles Through Homothety
  513. Twin Segments in Arbelos
  514. Two Butterflies Theorem
  515. Two Butterflies Theorem II
  516. Two Butterflies Theorem III
  517. Two Circles in a Parallelogram
  518. Two Circles Inscribed in a Parallelogram
  519. Two Circles in a Square
  520. Two Circles in a Square II
  521. Two Circles in an Angle
  522. Two Circles on a Side of a Triangle
  523. Two Circumcircles And Two Pairs of Parallels
  524. Two Circumcircles in Triangle
  525. Two Congruent Circles by Reflection
  526. Two Homotheties in a Parallelogram
  527. Two Proofs of MenelausTheorem
  528. Two Quadrilaterals
  529. Two Quadruplets of Tangent Circles
  530. Two Rectangles in a Rectangle
  531. Two Simsons in a Triangle
  532. Two Squares and Another Square
  533. Two Triples of Similar Triangles
  534. Two Tangents to Parabola
  535. Two Triangles With Common Base and Altitude
  536. Varignon Parallelogram
  537. Vecten's Collinearity
  538. Vecten's Mesh
  539. Vecten's Theorem
  540. Versatile Theorem
  541. Weitzenböck Inequality
  542. What does one see inside a spherical mirror?
  543. When a Triangle is Equilateral
  544. Wittenbauer's Parallelogram
  545. Y. Sawayama's Lemma
  546. Zaslavsky's theorem

    Miscellaneous Demonstrations

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


Copyright © 1996-2009 Alexander Bogomolny

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