Memory and Matching Arithmetic Algebra
Logic Fractals Probability
Visual Illusions Calculus Math Magic
Combinatorics Combinatorial Games Puzzles & Games
Geometric Fallacies Social Sciences Elementary Geometry
Stereometry Outline Mathematics Miscellaneous Demonstrations

Interactive Mathematics Activities

    Memory and Matching

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


  5. 1089 and a Property of 3-digit Numbers
  6. 2 Pails Puzzle
  7. 3 Jugs Puzzle
  8. 3 Jugs Puzzle in Barycentric Coordinates
  9. 3-Term Arithmetic Progression
  10. Abacus in Various Number Systems
  11. Algorithm for Computing LCM
  12. Aliquot Game
  13. Arithmetic Operations in Various Bases
  14. Broken Calculator
  15. Coin Game
  16. Collatz Conjecture
  17. Common Multiples and the Least Common Multiple
  18. Compare Fractions: Interactive Practice
  19. Counting and Grouping
  20. Distributive Law
  21. EAN-13 Encoding
  22. Equivalent Fractions
  23. Euclid's Algorithm: An Interactive Illustration
  24. Euclid's Game
  25. Euclid's Game on a Square Grid
  26. Even, Odd and Total Number of Digits
  27. Factoring with the Factor Tree
  28. Farey Fractions
  29. Fibonacci Numbers in Equilateral Triangle
  30. Flipping Items Simultaneously
  31. Fraction Comparison
  32. Fraction Comparison Sped up
  33. GCD and LCM via Factor Tree
  34. GCD and LCM by Plain Factorization
  35. GCD(M, N) × LCM(M, N) = M × N
  36. Geometric view of the Sieve of Eratosthenes
  37. ISBN Encoding
  38. Kaprekar's Iterations and Numbers
  39. Lattice Multiplication
  40. Long Multiplication
  41. Less, Equal, More
  42. Listing All the Composite Numbers
  43. Lucky Numbers
  44. Make An Identity
  45. Make It All Zeros
  46. Moving Chips in Pairs Down a Checkerboard
  47. Napier Bones
  48. Plain Multiplication table up to 20×20
  49. Product of Simple Fractions
  50. Rule of Change of Addends
  51. Sieve of Eratosthenes
  52. Sieve of Squares
  53. Soroban in Various Number Systems
  54. Suan Pan in Various Number Systems
  55. Subtraction in a Rectangular Array
  56. Sums and Products
  57. Sums of Powers of Digits
  58. Treasure Hunt in a Square Grid
  59. Treasure Hunt from Outside the Grid
  60. Two Simple Equations
  61. Universal Product Code
  62. Zeros and Nines


  63. A Line in a Square Grid
  64. ab+bc+ca ≤ aa+bb+cc
  65. Ambassadors at a Round Table
  66. An Arctan Identity and its Consequence
  67. Averages, Arithmetic and Harmonic Means
  68. An Inequality with Complex Numbers of Unit Length
  69. An Inequality with Complex Numbers of Unit Length II
  70. Base (Binary, Decimal, etc.) Converter
  71. Binary Color Device
  72. Breaking Chocolate Bars
  73. Candy Game: Integer Iterations on a Circle
  74. Candy Game (Change Discharged)
  75. Chebyshev polynomials
  76. Complex Numbers: A Dynamic Tool
  77. Constructible Numbers
  78. Counting Chips On a Circle
  79. Counting Squares in a Square
  80. Counting Triangles
  81. Counting Triangles II
  82. Divisibility Problem
  83. Dots and Fractions
  84. Ducci Sequences
  85. Egyptian Multiplication
  86. Filling an Array with 0s and 1s - and Counting
  87. First Geometric Interpretation of Complex Numbers
  88. Four Pegs That Form a Square
  89. Frieze Patterns
  90. Geometric Construction of Roots of Quadratic Equation
  91. Graph and Roots of a Third Degree Polynomial
  92. Graph and Roots of Quadratic Polynomial
  93. Graphing Equations Is Useful, II
  94. Graphing Equations Is Useful, V
  95. Heads and Tails (counting, invariance)
  96. Identities in the Multiplication Table
  97. Integer Iterations on a Circle II (superposition principle)
  98. Integer Iterations on a Circle III
  99. Integers and Rectangles
  100. Integers and Rectangles: a Proof by Induction
  101. Integers and Rectangles: Two Simple Proofs
  102. Interest Calculations
  103. Josephus Flavius Game
  104. Langford and Skolem Sequences
  105. Letter Count in a Sentence
  106. Lights Out
  107. Logistic Model
  108. Loop or Halt
  109. Mean Values
  110. Merlin's Magic Squares (modular arithmetic, boolean and linear algebra)
  111. Mini Lights Out
  112. Minimax Principle
  113. Minimax Theorem
  114. Modular Arithmetic
  115. Multiplication of Points on a Circle
  116. Multiplication of Points on an Ellipse
  117. Nine Squares
  118. Parrondo Paradox Via Simpson Paradox
  119. Peasant Multiplication
  120. Pennies in Boxes
  121. Plus or Minus (parity, invariance)
  122. PolygonalNumbers
  123. Proizvolov's Identity
  124. Pythagorean Triples
  125. Pythagorean Triples Calculator
  126. Quasigroup on a Triangular Grid
  127. Rectangle on a Chessboard
  128. Right Replacement
  129. Self-descriptive Strings
  130. Self-documenting Sentences
  131. Sierpinski's Gasket and Dihedral Symmetry
  132. Signs and Sums in a Sequence
  133. Signs in a Matrix
  134. Simple Cellular Automaton
  135. Simple Somos Sequence Calculator
  136. Simpson Paradox
  137. Sine of a Sum Formula
  138. Solitaire in Two Colors
  139. Squares and Circles (parity, invariance)
  140. Squares, Circles, and Triangles (modular arithmetic, invariance)
  141. Steinhaus' Problem
  142. Sum of Consecutive Integers is Triangular
  143. Sum of Consecutive Odd Numbers is Square
  144. Sum of Consecutive Triangular Numbers is Square
  145. Sums of Geometric Series - Proofs Without Words
  146. Swapping Rows and Columns
  147. Toads And Frogs Puzzle (counting, logic, problem solving)
  148. Two Colors in Two Rows
  149. Two Colors in Two Rows II
  150. Voronoi Diagram
  151. What's next?
  152. Word Problems


  153. Lewis Carroll's Bilateral Diagrams
  154. Lewis Carroll's Trilateral Diagrams
  155. Lewis Carroll's Logic Game
  156. Venn Diagrams I
  157. Venn Diagrams II
  158. Formal Systems - an Example


  159. All Peano Curves
  160. Apollonian Gasket
  161. Color Cycling on the Mandelbrot Set
  162. Cantor sets
  163. Dot Patterns, Pascal Triangle and Lucas Theorem
  164. Emergence of Chaos
  165. Exercise with Square Spiral
  166. Following the Hilbert Curve
  167. Fractal Curves and Dimension
  168. Iterated Function Systems
  169. Iterations in the Mandelbrot Set
  170. Koch's Snowflake
  171. Mandelbrot Set and Indexing of Julia Sets
  172. Plane Filling Curves: Hilbert's and Moore's
  173. Plane Filling Curves: One of Sierpinski's Curves
  174. Plane Filling Curves: the Lebesgue Curve
  175. Plane Filling Curves: Peano's and Wunderlich's
  176. Sierpinski Gasket By Common Trema Removal
  177. Sierpinski Gasket Via Chaos Game
  178. Sierpinski Gasket by Trema Removal
  179. The Chaos Game: Address Space vs IFS
  180. Variations on the Theme of Tremas
  181. Variations on the Theme of Tremas II


  182. Bertrand's Paradox
  183. Binomial Distribution
  184. Birds On a Wire
  185. Buffon's Noodle
  186. Chess Players Truel
  187. Conditional Probability and Independent Events
  188. Eight Selections in Six Sectors
  189. Family Statistics
  190. Lewis Carroll's pillow problem
  191. Monty Hall Dilemma's Simulation
  192. Parrondo Paradox
  193. Random Clock Hands
  194. Stick Broken Into Three Pieces (Cartesian Coordinates)
  195. Stick Broken Into Three Pieces (Trilinear Coordinates)
  196. Who Needs Monty?

    Visual Illusions

  197. Assimilation Illusion
  198. Balls and Shadows Illusion
  199. Brick Wall Illusion
  200. Bulging Lines illusion
  201. Circular Poggendorff Illusion
  202. Continuous Line Illusion
  203. Cube Counting Ambiguity
  204. Day's Sine Illusion
  205. Delboeuf Illusion
  206. Distorted Lines Illusion
  207. Divergent Spirals Illusion
  208. Gerbino's Illusion
  209. Heaver: A Visual Illusion
  210. Height/Width Interplay Illusion
  211. Hering Illusion
  212. Hermann Grid Illusion
  213. Importance of Occluders: Motion Binding
  214. Judd Illusion
  215. Kanizsa Triangle
  216. Müller-Lyer Illusion
  217. Necker Cube
  218. Occluded Wheel Illusion
  219. Orbison Illusion
  220. Parallelogram Illusion
  221. Perspective Illusion
  222. Poggendorff Illusion
  223. Poggendorff Illusion II
  224. Ponzo Illusion
  225. Ponzo Illusion II
  226. Reverse Spoke Illusion
  227. Revolving Circles Illusion
  228. Rotating Top Illusion
  229. Scintillating Grid Illusion
  230. Shepard's Parallelogram Illusion
  231. Shifting Cube Illusion
  232. Shimmer Illusion
  233. Simultaneous Contrast Illusion
  234. Simultaneous Contrast Illusion II
  235. Simultaneous Contrast: Koffka's Illusion
  236. Simultaneous Contrast: White's Illusion
  237. Simultaneous Contrast: White's Illusion, V-Variant
  238. Staircase Illusion
  239. Twisted Cord Illusion
  240. Unfolding Spirals Illusion
  241. Wundt Illusion
  242. Wundt Block Illusion
  243. Zöllner's Illusion


  244. Addition and Subtraction of Functions
  245. Area of a Circle by Leonardo da Vinci
  246. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
  247. Cartesian Coordinate System
  248. Cavalieri's Principle
  249. Derivative of Sine and Cosine
  250. Distance From a Point to a Straight Line
  251. Equations of a Straight Line
  252. Estimating Circumference of a Circle
  253. Focal Definition of Parabola
  254. Function, Derivative and Integral
  255. Graph of a Polynomial
  256. Graph of a Polynomial Defined by Its Roots
  257. Horner's Method
  258. Inflection Points of Fourth Degree Polynomials
  259. Lagrange Interpolation
  260. Linear Function with Coefficients in Arithmetic Progression
  261. Maximum Volume of a Cut Off Box
  262. Mistrust Intuition of the Infinite
  263. Naturally Discontinuous Functions
  264. Newton's Method
  265. Riemann Sums - Function Integration
  266. Rolle's and Mean Value Theorems
  267. Roots and Tangents
  268. Schwarz Lantern
  269. Taylor Series Approximation to Cosine
  270. Two Circles, Two Segments - One Ratio
  271. Two Circles and a Limit

    Math Magic

  272. An Arithmetic Magic Trick
  273. Arithmetic magic matrix
  274. Bachet's Magic Trick
  275. Barcode Magic
  276. Calendar Magic
  277. Gergonne's Magic Trick
  278. Hummer's Mind Reader
  279. Magic in Square
  280. Math Telepathy
  281. Number Guessing Game
  282. Two Numbers Guessing Game


  283. 3-Colors Tower of Hanoi
  284. Affirmative Action Problem
  285. Balanced Incomplete Block Design on a Triangular Grid
  286. Ballot Lemma
  287. Bicolor Tower of Hanoi
  288. Blithe 12
  289. Cheapest Link & Kruskal's Algorithms - skill practice
  290. Chvatal's Art Gallery Theorem
  291. Coin change
  292. Coloring a Graph
  293. Construction of Pascal's Triangle
  294. Critical Path in a Weighted Digraph
  295. Scheduling and Critical Path Algorithm
  296. de Bruijn Cycle
  297. Dominoes on a Chessboard
  298. Euler & Hamilton Paths - skill practice
  299. Filling a Grid with Good Neighbors
  300. Five Lattice Points
  301. Fleury's Algorithm and Euler's Paths and Cycles
  302. Flipping Items Simultaneously II
  303. Flipping Items Simultaneously III
  304. Flipping pancakes
  305. Futurama Theorem and Puzzle
  306. Gray Codes
  307. Group multiplication of permutations
  308. Finding Hamilton Paths and Circuits
  309. Hanoing
  310. Happy 8
  311. Hex Can't End in a Draw
  312. How to define a permutation
  313. Infinite Latin Square
  314. Jumping Isn't Everything
  315. Latin Squares
  316. Leibniz and Pascal Triangles
  317. Life-like Automaton With Definable Rules
  318. Light Bulbs in a Circle
  319. Listing All Combinations
  320. Listing All Permutations
  321. Listing All Permutations: Johnson-Trotter Method
  322. Lucky 7
  323. Make Your Move, Kid!
  324. Marriage Problem, simulation
  325. Mazes
  326. Minumum Spanning Trees and Kruskal's Algorithm
  327. Monotone Number Sequences
  328. No Flipping without End
  329. Number sequences and Pigeonhole
  330. Party Acquaintances
  331. Permutations as a Product of Transpositions
  332. Product of Permutations
  333. Project Scheduling
  334. Property of the Powers of 2
  335. Random Walks
  336. Recombining Permutations
  337. Reflection Lemma
  338. Sam Loyd's Fifteen
  339. Scrub Tile Puzzle
  340. Shuttle Puzzle
  341. Simple Graphs
  342. Simple Graph Practice
  343. Sliders (slider games on torus, Moebius strip, projective plane, permutations, cycles)
  344. Straight Tromino on a Chessboard
  345. The Game of Life
  346. Tiling Rectangles with L-Trominoes
  347. Tool for Constructing Orthogonal Latin Squares
  348. Tower of Hanoi
  349. Tower of Hanoi, the Hard Way
  350. Traveling Salesman Problem
  351. Two Men of Tibet
  352. Universal Coloring
  353. Y Can't End in a Draw

    Combinatorial Games

  354. Date Game
  355. Dawson's Chess
  356. Dawson's Kayles
  357. Fraction Game
  358. Grundy's Game
  359. Hex 7
  360. Kayles
  361. Nim
  362. Nimble
  363. Northcott's game
  364. Odd Scoring
  365. One Pile
  366. Plainim
  367. Plainim Misère
  368. Scoring
  369. Scoring Misère
  370. Scoring Misère: Two Heaps Perfect Strategy
  371. Silver Dollar Game
  372. Silver Dollar Game With No Silver Dollar
  373. Sticks, Jess realization of Scoring Misère
  374. Sticky Problem
  375. Sticky Problem II
  376. Subtraction Game
  377. TacTix
  378. Take-Away Games
  379. Turning Turtles
  380. Wythoff's Nim
  381. Wythoff's Nim II

    Puzzles & Games

  382. 3-Chain
  383. 4×4×4 Tic Tac Toe (spacial visualization)
  384. A Game of Candy Squares
  385. Achieve the Equilibrium
  386. Bulls and Cows
  387. Caterpillar
  388. Changing Colors (combinatorics, invariance)
  389. Changing Colors II (invariance)
  390. Color Balls
  391. Computer Mastermind (logic and problem solving)
  392. Covering a Chessboard with a Hole with L-Trominoes
  393. Eight Digits Puzzle
  394. Escape of the Clones
  395. Farmer and Wife To Catch Rooster and Hen (invariance, city block distance)
  396. Fif (multiple representations)
  397. Fill a Hexagonal Grid (space visualization)
  398. Fish Soup Game (multiple representations)
  399. Four Knights (multiple representations)
  400. Full House
  401. Getting a Scout out of Desert
  402. Goat, Cabbage and Wolf to move across a river (multiple representations)
  403. Hare and Hounds
  404. Jam (multiple representations)
  405. Looping Chips (multiple representations)
  406. Mastermind (logic and problem solving)
  407. Mastermind Variants (logic and problem solving)
  408. Mirror Maze
  409. Peg Solitaire (logic, problem solving, subproblems)
  410. Pick A Label (logic, problem solving)
  411. Reverse Solitaire (logic, problem solving, subproblems)
  412. Sending scouts into the desert (algebra, geometric series, problem solving)
  413. Seven Coins Puzzle (multiple representations)
  414. Shuttle Puzzle Practice I
  415. Shuttle Puzzle Practice II
  416. Skyscrapers Puzzle
  417. Solitaire on a Circle (iterations, invariance)
  418. Stones in Cups
  419. Super Nim
  420. Thugs and Bowmen
  421. Tiling a Triangulated Hexagon (a proof without words)
  422. Toads And Frogs in 2 dimensions (repeated application of same idea)
  423. Tromino Puzzle: Golomb's Theorem
  424. Tromino Puzzle: Deficient Squares
  425. Vector Addition and Subtraction
  426. Vowel Movement
  427. Wolf and Rabbit

    Social Science

  428. Adams' Apportionment Method
  429. Banzhaf Power Index Calculator
  430. Fair Division: Method of Lone Divider
  431. Fair Division: Method of Markers
  432. Fair Division: Method of Sealed Bids
  433. Fair Division: Method of Sealed Bids II
  434. Fast Power Indices
  435. Five Methods of Apportionment
  436. Four Voting Methods
  437. Hamilton's Apportionment Method
  438. Huntington-Hill Apportionment Method
  439. Jefferson's Apportionment Method
  440. Method of Markers II
  441. Shapley-Shubik Power Index Calculator
  442. Social Choice
  443. Webster's Apportionment Method
  444. Weighted Voting and Power Indices

    Elementary Geometry

  445. 5x5 Square Grid and 5 Circes
  446. Puzzle of 30 Grid Points and 5 Circles
  447. A Characterization of the Euler Line
  448. A Degenerate Case of Bottema's Configuration
  449. A Diameter As a Diagonal of Inscribed Quadrilateral
  450. A Double Meaning of an Arc's Midpoint
  451. A Family of Cyclic Quadrilaterals
  452. A Hinged Realization of a Plane Tessellation
  453. A Lemma of Equal Areas
  454. A Median Surprise
  455. A Multiplicative Identity of Areas in a Triangle
  456. A Property of a Median
  457. A Property of Equiangular Polygons
  458. A Property of Isogonal Lines
  459. A Property of Orthic Triangle
  460. A Property of Perspective Triangles
  461. A Three Pegs Question
  462. A Triangle With a 45 Degrees Angle in Square
  463. Acute Angle Bisectors in a Right Triangle
  464. Al-Nayrizi's Construction
  465. An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem
  466. An Extra Triple of Equilateral Triangles for Napoleon
  467. An Unexpected Rhombus
  468. Angle: An Illustrated Classification
  469. Angle Bisector
  470. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise
  471. Angle Bisectors in Ellipse
  472. Angle Bisectors in Ellipse II
  473. Angle Bisectors On Circumcircle
  474. Angle Subtended by a Diameter
  475. Angles Inscribed in an Absent Circle
  476. Anticenter and Orthocenters
  477. Antiparallel
  478. Antiparallel via Three Reflections
  479. Archimedes' Law of the Lever
  480. Area of Median Triangle I
  481. Area of Median Triangle II
  482. Area of Parallelogram
  483. Area of the Intersection of Two Equal Rectangles
  484. Area of the Union of Two Squares
  485. Area of Triangle
  486. Areas between Two Parallelograms
  487. Areas in Square by Dissection
  488. Areas in Two Parallelograms and a Translation
  489. Arithmetic in Disguise
  490. Bespectacled Eyeballs Extension
  491. Between Major and Minor Circles
  492. Bottema with a Product of Rotations
  493. Bottema with Similar Triangles
  494. Bottema's Theorem - Proof Without Words
  495. Building a Bridge
  496. Building Bridges
  497. Carpet With a Hole
  498. Carpets in Triangle
  499. Center of Gravity and Motion of Material Points
  500. Chain of Four Intersecting Circles
  501. Characterization of Trapezoid
  502. Circle and Hyperbola as Lighthouse Curves II
  503. Circle Chains on Napoleon Triangles
  504. Circle Rotations and Fixed Points
  505. Circles and Semicircles in Rectangle
  506. Circle in a Square Inscribed in a Circle
  507. Circles with Equal Collinear Chords
  508. Circumcenter and Orthocenter
  509. Circumcenter on Angle Bisector
  510. Classification of Quadrilaterals
  511. Common Chord and a Tangent
  512. Concurrence in Arbelos
  513. Concurrence of Ten Nine-Point Circles
  514. Concurrence on a Circle
  515. Concurrent and Parallel Lines in Parallelogram
  516. Concyclicity from Collinearity
  517. Concyclicity in Rectangle
  518. Compass Only Construction - A Chord Tangent to an Inner circle
  519. Concyclic Points from Midpoint of an Arc
  520. Concyclicity in Two Semicircles
  521. Conic from Parallel Chords
  522. Conic in Hexagon
  523. Conic Sections as Loci of Points
  524. Conics in Triangle
  525. Construct Triangle by Angle, Altitude and Median
  526. Construct Triangle by Angle, Median, and Circumradius
  527. Construction: Parallel Line through Point
  528. Construction of a Right Triangle
  529. Construction of an Oval - the Roman Way
  530. Construction of the Angle Bisector
  531. Construction of the Perpendicular Bisector
  532. Cross Points in a Polygon
  533. Cutting a Hole in Half
  534. Cyclic Quadrilateral, Angle Bisector And Isosceles Triangle
  535. Dancing Rectangles Model Auxetic Behavior
  536. Dancing Squares or a Hinged Plane Tessellation
  537. A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
  538. Diagonal Count
  539. Dissection of a Vase
  540. Dissection of Cyclic Quadrilateral
  541. Divide a Circle into N Parts of Equal Area
  542. E. W. Dijkstra's Proof of the Pythagorean Theorem
  543. Eight Equal Tangents
  544. Equal Areas in Equilateral Triangle
  545. Equal Areas in Regular 2n-gons
  546. Equal Chords in Crossing Circles
  547. Equal Circles, Medial Triangle and Orthocenter
  548. Equidecomposition of a Rectangle and a Square
  549. Equidecomposition of Two Rectangles
  550. Equidecomposition of a Triangle and a Rectangle
  551. Equidecomposition of a Triangle and a Rectangle II
  552. Equidecomposition of Two Parallelograms
  553. Equilateral Triangle on a Closed Curve
  554. Equilateral Triangle on Parallel Lines
  555. Equilateral Triangle on Parallel Lines II
  556. Equilateral Triangle on Three Lines
  557. Equilateral Triangles formed by Circumcenters
  558. Equilateral Triangle in Equilateral Triangle
  559. Equilic Quadrilateral I
  560. Equilic Quadrilateral I, A Variation
  561. Equilic Quadrilateral II
  562. Euclid's Segment Division
  563. Existence of the Circumcenter
  564. Existence of the Euler Line: An Elementary Proof
  565. Existence of the Orthocenter
  566. Extras in Bottema's Configuration
  567. Eye-to-Eye Theorem I
  568. Eye-to-Eye Theorem II
  569. Fibonacci Bamboozlement
  570. Famous Theorem
  571. Finding Circle Center By Ruler Alone
  572. Five Squares in Complex Numbers
  573. Fixed Point in Isosceles and Equilateral Triangles
  574. Fixed Point on Perpendicular Bisector
  575. Focal Definition of Ellipse
  576. Focus and Directrix of Ellipse
  577. Folding Square in a Line through the Center
  578. Four Circles and a Line
  579. Four Hinged Squares
  580. From Arbitrary Hexagon to Regular One
  581. From Perpendicular Center Lines to Concyclic Points
  582. From Straight Line to Regular Hexagon
  583. From Arbitrary Pentagon to Regular One
  584. Garcia's Two Circles Lemmas
  585. Geoboard
  586. Half Turn, Reflection in Point
  587. Harmonic Mean Sangaku
  588. Hinged Greek Cross Tessellation
  589. Hippocrates' Squaring of a Lune
  590. Homothety in Three Tangent Circles
  591. Homothety in Three Tangent Circles II
  592. How to Construct Tangents from a Point to a Circle
  593. Hunting Right Angles
  594. Huygens' proof of the Pythagorean Theorem
  595. In the Spirit of Thebault I
  596. Inscribed and Central Angles in a Circle
  597. Inscribed Angles
  598. Inscriptible Quadrilateral: An Illustration
  599. Integer Triangles with Two Angles in the Ratio 1 : 2
  600. Intersecting Chords Theorem - a Visual Proof
  601. Intersecting Chords Theorem - Hubert Shutrick's PWW
  602. Intersections of a Circle with the Four Quadrants
  603. Invariance in Orthodiagonal Quadrilaterals
  604. Is X a Midpoint of a Chord
  605. Isogonal Lemma
  606. Isogonal Concurrencies
  607. Isoperimetric Theorem For Quadrilaterals
  608. Isoperimetric Theorem For Quadrilaterals II
  609. Isosceles Configuration in Triangle
  610. Join Circles by Given Segment
  611. Joined Common Chords of Napoleon's Circumcircles
  612. Law of Cosines - Another PWW
  613. Lean Napoleon's Triangles
  614. Line Not Through a Center of Similarity
  615. Line Through a Center of Similarity
  616. Looking Back at Bottema
  617. Maimed Cake
  618. Mathematics in Pizzeria
  619. Midline in Trapezoid
  620. Midline in Triangle
  621. Miguel's Area of Square
  622. Miguel Ochoa's Chords And Tangents
  623. Miguel Ochoa's van Schooten is a Slanted Viviani
  624. Miguel Ochoa's van Schooten Like Theorem
  625. Miguel Ochoa's van Schooten Like Theorem II
  626. Miguel Ochoa's van Schooten Like Theorem III
  627. Miquel Circumcenters
  628. Napoleon's Hexagon
  629. Napoleon's Theorem: A third proof with complex numbers
  630. Napoleon's Theorem via Two Rotations
  631. Napoleon on Hinges
  632. Napoleon on Hinges in GeoGebra
  633. Neuberg Cubic
  634. Nine Point Circle: an Elementary Proof
  635. Octagon in Parallelogram
  636. Octagon in Parallelogram: Overlapping Regions
  637. Olof Hanner's Jigsaw Puzzle
  638. Orthic Semiperimeter
  639. Orthocenter and Circumcenter Are Isogonal Conjugate
  640. Orthocenters of Two Triangles Sharing Circumcenter and Base
  641. Orthodiagonal and Cyclic Quadrilaterals
  642. Orthogonality in Two Squares
  643. Pappus' Generalization of Euclid I.47
  644. Parabolic Reciprocity
  645. Parallel Chords II
  646. Parallel Chords in Crossing Circles
  647. Parallel Lines in a Quadrilateral
  648. Parallel Lines in a Quadrilateral II
  649. Parallelogram and Four Equilateral Triangles
  650. Parallelograms among Quadrilaterals
  651. Parallelogram and Ellipses
  652. Parallelogram in Four Circles
  653. Parallelogram in Parallelogram And in Projection
  654. Parallelogram with Side Lines through Fixed Points
  655. Pascal: Necessary and Sufficient
  656. Pedal Similarity
  657. Pedal Triangles of Inverse Points
  658. Perimeters of Convex Polygons, One within the Other
  659. PLL: Apollonius' Problem with Two Lines and a Point
  660. Polygon Metamorphosis
  661. Pompeiu's Theorem
  662. Praying Eyes Theorem
  663. Problem 1 from the EGMO2017
  664. Proclivity of Conjoint Archimedean Twins for Proliferation
  665. Properties of Flank Triangles - Second Proof by Symmetric Rotation
  666. Properties of Flank Triangles - Proof with an Asymetric Rotation
  667. Property of Angle Bisectors
  668. Property of Angle Bisectors III
  669. Property of Internal Angle Bisector - Hubert Shutrick's PWW
  670. Property of Rhombi
  671. Property of Right Isosceles Triangle
  672. Property of Semicircles
  673. Pure Angle Chasing
  674. Pure Angle Chasing II
  675. Pure Angle Chasing III
  676. Pythagorean Theorem By Dissection
  677. Pythagorean Theorem By Dissection II
  678. Pythagorean Theorem by Hinged Dissection
  679. Pythagorean Theorem by Hinged Dissection I
  680. Pythagorean Theorem by Hinged Dissection II
  681. Pythagorean Theorem: J. Adams' Proof by Dissection
  682. Pythagorean Theorem: Versluys' Proof by Dissection
  683. Pythagoras' Theorem By Sheer Shearing
  684. Pythagoras' Theorem By Similarity
  685. Pythagoras' Theorem Solely By Sliding
  686. Pythagorean Theorem via an Isosceles Triangle
  687. Pythagorean Segment Division
  688. Quadrilateral With Equal Opposite Sides And Angles
  689. Rectangles among Parallelograms
  690. Rectangular Areas in Circle
  691. Reflection in the Circumcenter
  692. Regular Hexagon and Triangle Joined at Vertex
  693. Reim's Similar Coins I
  694. Reim's Similar Coins II
  695. Reim's Similar Coins III
  696. Reim's Similar Coins IV
  697. Rhombus in a Cyclic Quadrilateral
  698. Right Triangles on Sides of a Square
  699. Rhombus in Circles
  700. Sangaku with Angle between a Tangent and a Chord
  701. Secant Angles
  702. Secant Angles II
  703. Secant, Normal, Tangent
  704. Segment Trisection II
  705. Segment Trisection Induced by Parallels to Medians
  706. Semi-regular Tessellation on Hinges A
  707. Semi-regular Tessellation on Hinges B
  708. Semi-regular Tessellation on Hinges C
  709. Seven Concyclic Points
  710. Shearing a Polygon into a Triangle of Equal Area
  711. Shifted by the Inradius Tangent Collinearities
  712. Similar Triangles in Crossing Circles
  713. Simple Property of Circle Through the Incenter
  714. Simple Quadrilaterals Tessellate the Plane
  715. Shearing and Translation in Pythagorean Pants
  716. Some Properties of Common Tangents
  717. Spiral Similarity Leads to Equilateral Triangle
  718. Square, Semicircle, and Other Curves
  719. Square From Nowhere
  720. Square in a Circle Inscribed in a Square
  721. Square in Chair
  722. Squaring a Rectangle
  723. Sum of Two Squares
  724. Symmetries in Triangle II
  725. Symmetry in an Asymmetric Configuration
  726. Tangent Circles, Similarity and Equal Segments
  727. Tangent Curiosity in Equilateral Triangle
  728. Thales' Theorem
  729. Thales on Angle Bisectors
  730. The Eutrigon Theorem
  731. The Shape of the Colosseum
  732. Three Collinear and Four Concyclic Points
  733. Three Common Chords in Three Concurrent Circles
  734. Three lines at 60 degres and Circumcenters
  735. Three Touching Circles
  736. Translated Triangles
  737. Translation Transform
  738. Triangle Areas in a Parallelogram
  739. Triangle Areas in a Parallelogram II
  740. Triangle Classification
  741. Trigonometric Functions
  742. Triangular Areas in Circle
  743. Twin Segments in Four Semicircles
  744. Two Altitudes, One Midpoint
  745. Two Color Coloring of the Plane
  746. Two Common Chords in Three Concurrent Circles
  747. Two Diameters and Longest Common Chord
  748. Two Equilateral Triangles
  749. Two Intersecting Circles
  750. Two Parallels in a Triangle and One More
  751. Two Pascals Merge into One
  752. Two Properties of Flank Triangles - First Proof by Symmetric Rotation
  753. Two Properties of Flank Triangles - A Proof with Complex Numbers
  754. Two Properties of Flank Triangles - and a Third One
  755. Two Regular Pentagons Joined at Vertex
  756. Two Symmetric Triangles Are Directly Equidecomposable
  757. Two Symmetric Triangles Are Directly Equidecomposable II
  758. Two Symmetric Triangles Are Directly Equidecomposable III
  759. Two Symmetric Triangles Are Directly Equidecomposable IV
  760. Two Touching Circles
  761. Van Schooten's and Pompeiu's Theorems
  762. Viviani's Theorem
  763. Viviani's Theorem II
  764. Viviani in Isosceles Triangle
  765. Yet Another Generalization of Bottema's Theorem


  766. Angles in a Cube I
  767. Angles in a Cube II
  768. Angles in a Cube III
  769. Bimedians in a Regular Tetrahedron
  770. Cut the Cone
  771. Cut the Cube
  772. Cut the Cylinder
  773. Knots on a Torus
  774. Möbius' Strip
  775. Multifaceted Cork
  776. One Sheet Hyperboloid
  777. Platonic Solids - Regular Polyhedra
  778. Other Solids


    Geometric Fallacies

  1. A Circle With Two Centers
  2. All Triangles Are Isosceles
  3. Curry's Paradox
  4. Delian Problem Solved
  5. A Faulty Dissection
  6. Hooper's Paradox
  7. Is Every Trapezoid Parallelogram?
  8. John Sharp's Paradox
  9. Langman's Paradox
  10. Popping A Square
  11. Rouse Ball's Fallacy
  12. Sam Loyd's Son's Dissection
  13. Two Perpendiculars From a Point to a Line

    Outline (Fill-It-In) Mathematics

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


  15. 120° Breeds 90°
  16. 2N-Wing Butterfly Theorem
  17. 3 Isosceles Trapezoids
  18. 5-Star and A Circle
  19. 6 to 9 Point Circle
  20. 60° Breeds 90°
  21. 9-point Circle as a Locus of Concurrency
  22. 9 Point Center on Angle Bisector
  23. A Chain of Touching Circles in a Polygon (à la Quang Tuan)
  24. A Chain of Touching Circles in a Polygon (à la Evelyn)
  25. A Circle Related to Incenter and Circumcenter
  26. A Circle Rolling in an Equilateral Triangle
  27. A Conic in a Triangle
  28. A Generalization of Simson Line
  29. A Geometry Problem from 2012 IMO - Problem 5
  30. A Line in Triangle Through the Circumcenter
  31. A Parallelogram in Triangle
  32. A Problem in Three Pentagons
  33. A Sangaku Follow-Up on an Archimedes' Lemma
  34. A Sangaku: Two Unrelated Circles
  35. A Sangaku with an Egyptian Attachment
  36. A Sangaku with Many Circles and Some
  37. A 60° Rhombus in a 60° Isosceles Trapezoid
  38. A Stronger Triangle Inequality
  39. A Triangle of Antreas Hatzipolakis
  40. Abutting Equilateral Triangles on a Line I
  41. Adams' Circle
  42. Addition of Radius-Vectors
  43. Altitudes and the Power of a Point
  44. An Euclidean Construction with Inversion
  45. An Isoperimetric Theorem
  46. An Unexpected Pair of Similar Triangles Which Are Equal
  47. Angle Bisector in Equilateral Trapezoid
  48. Angle Bisector in Parallelogram
  49. Angle Bisector in Touching Circles
  50. Angle Bisectors and Perpendiculars in a Quadrilateral
  51. Angle Bisectors in a Quadrilateral
  52. Angle Preservation Property
  53. Angle Trisection by Archimedes
  54. Angle Trisection by Hippocrates
  55. Angle Trisection by Paul Vjecsner
  56. Angle Trisectors on Circumcircle
  57. Angles in Triangle Add to 180°
  58. Another Concurrence on the 9-Point Circle
  59. Another Geometry Problem from 2012 IMO - Problem 1
  60. Another Equilateral Triangle in Napoleon's Configuration
  61. Another Pair of Twins in an Arbelos
  62. Another Property of the 9-Point Circle
  63. Another Seven Circles Theorem
  64. Anticomplementary Triangle Surprise
  65. Antiparallel and Circumradius
  66. Apollonian Circle for Two Lines and a Circle
  67. Apollonian Circles Theorem
  68. Apollonius Problem
  69. Application of Ceva's Theorem
  70. Archimedean Siblings out of Wedlock, i.e., Arbelos
  71. Archimedes' Quadruplets
  72. Archimedes Triangle and Squaring of Parabola
  73. Archimedes' Twin Circles and a Brother
  74. Area of an Arbelos
  75. Areas and Centroid in a Triangle
  76. Areas in Three Squares
  77. Areas in Triangle
  78. Areas in Triangle II
  79. Arithmetic Mean Sangaku
  80. Arithmetic Mean Sangaku II
  81. Around the Incircle
  82. Around the Incircle in a Polygon
  83. Asian Pacific Kite
  84. Asymmetric Propeller I
  85. Asymmetric Propeller II
  86. Asymmetric Propeller III
  87. Asymmetric Propeller by Plane Tiling
  88. Barycentric Coordinates: a Tool
  89. Batman's Problem
  90. Bevan's Point and Theorem
  91. Bisectal Circle
  92. BMO 2014, Problem 3
  93. Book of Lemmas: Proposition 1
  94. Book of Lemmas: Proposition 2
  95. Book of Lemmas: Proposition 3
  96. Book of Lemmas: Proposition 5
  97. Book of Lemmas: Proposition 6
  98. Book of Lemmas: Proposition 9
  99. Book of Lemmas: Proposition 10
  100. Book of Lemmas: Proposition 12
  101. Book of Lemmas: Proposition 13
  102. Böttcher's Proof of the Pythagorean Theorem
  103. Böttcher's Proof Variant
  104. Bottema Shatters Japan's Seclusion
  105. Bottema's Theorem
  106. Bottles in a Slanted Rack
  107. Bottles in a Wine Rack
  108. Bottles in a Wine Rack: Proofs and Generalizations
  109. Brahmagupta's Theorem
  110. Brianchon in Ellipse
  111. Brianchon's Theorem
  112. Bride's Chair
  113. Broken Chord Theorem
  114. Broken Chord Theorem by Paper Folding
  115. Broken Chord Theorem: G. Patruno's proof
  116. Broken Chord Theorem: S. Anderson's proof
  117. Butterflies in a Pencil of Conics
  118. Butterfly in Ellipse
  119. Butterfly in Parabola
  120. Butterfly Problem
  121. Cabart's Collinearity
  122. Cantor's Theorem
  123. Cardioid As Envelope of Circles
  124. Cardioid As Epicycloid
  125. Carnot's Theorem
  126. Carpets in a Parallelogram
  127. Carpets in a Quadrilateral
  128. Carpets in a Quadrilateral II
  129. Carpets Theorem
  130. Carpets Theorem With Parallelograms
  131. Casey's Theorem
  132. Cassini's Ovals and Geometric Optimization
  133. CCC: Apollonius' Problem with Three Intersecting Circles
  134. CCC: Gergonnes's Solution to Apollonius' Problem
  135. Center Circles and Their Chains
  136. Centroids in a Polygon
  137. Ceva and Menelaus Theorems for Angle Bisectors
  138. Ceva's Theorem
  139. Ceva's Theorem (PWW)
  140. Cevian Cradle
  141. Cevian Cradle II
  142. Cevian Nest
  143. Cevian Parallelogram
  144. Cevian Triangle
  145. Chain of Circles on a Chord
  146. Chain of Inscribed Circles
  147. Chain of Six Intersecting Circles
  148. Chasing Angles in Pascal's Hexagon
  149. Chasles' Theorem, a Proof
  150. Cherchez le quadrilatère cyclique
  151. Cherchez le quadrilatère cyclique II
  152. Chords, Concurrency and Orthic Triangle
  153. Chords in Cardioid
  154. Circle and Hyperbola as Lighthouse Curves
  155. Circle Centers on Radical Axes
  156. Circle Chains and Inscribed Angles
  157. Circle Concurrence on Circumcircle/a>
  158. Circle Concurrency and Spiral Similarity
  159. Circle Crosses Triangle
  160. Circle Division by Euclid and Leonardo
  161. Circle Inscribed in a Circular Segment
  162. Circle of Similitude
  163. Circle through the Circumcenter
  164. Circle through the Incenter
  165. Circle through the Incenter And Antiparallels
  166. Circle-Stacking Theorem
  167. Circle, Isosceles Triangle and Fixed Point
  168. Circles And Parallels
  169. Circles and Reflections in a Triangle
  170. Circles Cover a Quadrilateral
  171. Circles in a Regular Polygon
  172. Circles in Morley's Triangles
  173. Circles Lined on the Legs of a Right Triangle
  174. Circles On Cevians
  175. Circles Tangent to Circumcircle
  176. Circles Tangent to Medians at the Centroid
  177. Circles through the Orthocenter
  178. Circles with Tangent Diameters
  179. Circumcevian Triangle
  180. Cleavance Center
  181. Cleaver
  182. Clifford Chain
  183. Clifford's Lemma
  184. Coaxal Circles on Perpendicular Bisector
  185. Coaxal Circles Theorem
  186. Colburn's Proof of the Pythagorean Theorem
  187. Collinear Intersections and Products of Ratios
  188. Collinearity from the IMO 2013 (Problem 4)
  189. Collinearity via Concyclicity
  190. Collinearity with the Orthocenter
  191. Common Centroids Lead to Equilateral Triangle
  192. Common Tangents to Two Circles I
  193. Common Tangents to Two Circles II
  194. Complete Quadrilateral
  195. Concurrence in Equilateral Triangle
  196. Concurrence in Right Triangle
  197. Concurrence Not from School Geometry
  198. Concurrence of the Altitudes As Seen from 3D
  199. Concurrency in Analytic Geometry
  200. Concurrency in Arbelos
  201. Concurrency in the Intouch Triangle
  202. Concurrent Cevians and a Conic through Their Feet
  203. Concyclic Circumcenters: A Dynamic View
  204. Concyclic Points in a Triangle
  205. Concyclic Incenters in Bicentric Quadrilateral
  206. Concyclic Points at the IMO 2014
  207. Concyclic Points in Bride's Chair
  208. Concyclic Points in Equilateral Bumps
  209. Concyclic Points in Inscriptible Quadrilateral
  210. Concyclic Points of Two Ellipses with Orthogonal Axes
  211. Conic in Mixtilinear Incircles
  212. Conjugate Diameters in Ellipse
  213. Construct Triangle by Angle Bisector, Altitude, and Side
  214. Construction and Properties of Mixtilinear Incircles
  215. Construction and Properties of Mixtilinear Incircles 2
  216. Construction of Conics from Pascal's Theorem
  217. Construction of Thebault Circles
  218. Consecutive Isosceles Decomposition
  219. Conway's Circle
  220. Copernicus' Theorem
  221. Crossed-Lines Construction of Shapes of Constant Width
  222. Cutting Triangle in Two by a Line through a Point
  223. Cyclic Hexagon
  224. Cyclic Incenters
  225. Cyclic Quadrilateral, Concurrent Circles and Collinear Points
  226. Cycloids
  227. Dao's Archimedean Twins
  228. Dao's Variant of Thébault's First Problem
  229. David Richeson's Extention of an Old Japanses Theorem
  230. Diameters and Chords
  231. Directly Similar Figures
  232. Desargues' Hexagon
  233. Desargues' Theorem
  234. Discovery of Duane DeTemple
  235. Dissection of Triangle into Rhombus
  236. Dissection of Triangle into Rhombus by Hubert Shutrick
  237. Divide Triangle by Lines Parallel to Base
  238. Dividing a Segment into N parts: Besteman's Construction
  239. Dividing a Segment into N parts: Besteman's Construction II
  240. Dividing a Segment into N parts: GLaD Construction
  241. Dividing a Segment into N parts: SaRD Construction
  242. Dividing Evenly a Quadrilateral
  243. Dividing Evenly a Quadrilateral II
  244. Double Generation Theorem
  245. Douglas' Theorem (complex variables, linear algebra)
  246. Droz-Farny Circles
  247. Droz-Farny Line Theorem
  248. Droz-Farny Theorem - an Analytic Solution
  249. Dual of Maxwell's Theorem
  250. Dual to Pappus' Theorem
  251. Dynamic Construction of Ellipse and Other Curves
  252. Dynamic Software as Serendipity Enhancement
  253. Easy Construction of Bicentric Quadrilateral
  254. Easy Construction of Bicentric Quadrilateral II
  255. Eight Point Circle
  256. Ellipse, how to draw
  257. Ellipse Between Two Circles
  258. Ellipse in Arbelos
  259. Ellipse Touching Sides of Triangle at Midpoints
  260. >Emil Stoyanov's New Year's Problem
  261. Equal Angles in Two Circles
  262. Equal Angles in Two Circles II
  263. Equal Areas In Parallelogram à la Pythagoras
  264. Equal Incircles Theorem
  265. Equichordal Curves
  266. Equilateral and 3-4-5 Triangles
  267. Equilateral Triangle, Straight Line and Tangent Circles
  268. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  269. Equilateral Triangles on Segments of Equilic Quadrilateral
  270. Equilateral Triangles On Sides of a Parallelogram
  271. Equilateral Triangles On Sides of a Parallelogram II
  272. Equilateral Triangles on Sides of a Parallelogram III
  273. Equilateral Triangles on Sides of a Quadrilateral
  274. Escher's Theorem
  275. Euler Line Cuts Off Equilateral Triangle
  276. Euclid I.43
  277. Euclid I.43 Extended
  278. Euclidean Construction of Center of Ellipse
  279. Euclidean Construction of Tangent to Ellipse
  280. Excircles in Trapezoid
  281. Exeter Point
  282. Extouch Triangle in Poncelet Porism
  283. Extra Feature of Van Aubel Configuration
  284. Extremal Distance Ratio
  285. Eyeball Theorem
  286. Eyeballing a Ball
  287. Eyeballs Projected
  288. External Angle Bisector
  289. Fagnano's problem
  290. Fagnano's problem II
  291. Fagnano's Problem, Third Solution
  292. Fagnano's Problem, Fourth Solution
  293. Fagnano's Problem in Reverse
  294. Fermat's Hexagon
  295. Fermat point and 9-point Centers
  296. Fermat's Point
  297. Fermat Points and Concurrent Euler Lines I
  298. Fermat Points and Concurrent Euler Lines II
  299. Feuerbach's theorem
  300. Feuerbach's Theorem: A Proof
  301. Final Chapter of the Asymmetric Propeller Story
  302. Find a Common Chord of Given Length
  303. Find the Center of a Given Circle
  304. First Vecten Point
  305. Five Cardioids
  306. Five Concyclic Points
  307. Five Incircles in a Square
  308. Five Incircles Theorem
  309. Fixed Point of Circles Orthogonal to the Given One
  310. Fixed point on a rod rotating around a triangle
  311. Focal Properties of Parabola
  312. Focus on the Eyeball Theorem
  313. Foot of Altitude and Minimum Distance
  314. For Equality Choose Angle Bisector
  315. Four 9-Point Circles in a Quadrilateral
  316. Four Centroids and Parallels
  317. Four Circles In a Triangle
  318. Four Concurrent Lines in a Cyclic Quadrilateral
  319. Four Crossing Circles
  320. Four Incident Circles
  321. Four Incircles in Equilateral Triangle
  322. Four Isosceles Triangles Hinged at Vertices
  323. Four Pedal Circles
  324. Four Touching Circles
  325. Four Touching Circles II
  326. Four Touching Circles III
  327. The Four Touching Circles Problem IV
  328. Four Triangles, One Circle
  329. Four Turtles (or Bugs, or Dogs, etc.)
  330. François Viète's Reduction of CCC to PCC
  331. Friendly Kiepert's Perspectors
  332. From Foci to a Tangent in Ellipse
  333. Fundamental Theorem of 3-Bar Motion
  334. Further properties of Van Aubel Configuration
  335. Fuss' Theorem
  336. Garcia's Archimedean Quadruplets
  337. Garcia-Feuerbach Collinearity
  338. Generalizing a Romanian Olympiad Problem
  339. Geometric Optimization from the Asian Pacific Mathematical Olympiad
  340. Gergonne and Medial Triangles Are Orthologic
  341. Gergonne and Soddy Lines Are Perpendicular
  342. Gergonne in Ellipse
  343. Glide Reflection
  344. Golden Ratio in an Irregular Pentagon, Construction I
  345. Golden Ratio in an Irregular Pentagon, Construction II
  346. Golomb's inductive proof of a tromino theorem
  347. Gothic Arc
  348. Grebe: from Ladies' Diary to Carroll's Pillow
  349. Grebe's Theorem
  350. Greg Markowsky's Problem for Parabola
  351. Griffiths Points in Cyclic Quadrilateral
  352. Harmonic Ratio
  353. Hart Circle
  354. Hart's Inversor
  355. Haruki's Lemma
  356. Heron's Formula: a Proof
  357. Heron's Problem
  358. Heron in Trapezoid
  359. Hexagon Parallel to Medial Triangle
  360. Hexagon Parallel to Orthic Triangle
  361. Hidden Orthogonality
  362. Hinged Squares
  363. Hinged Squares II
  364. Hjelmslev Theorem
  365. Hjelmslev's Theorem on Canvas
  366. Hofstadter Triangles and Points
  367. Homologous Lines under Three Spiral Similarities
  368. Homothety
  369. Homothety between In- and Excircles
  370. Homothety in Equilateral Triangles
  371. How to Construct a Radical Axis
  372. How to Construct Orthogonal Circle
  373. Hypocycloid Families
  374. Ian McGee's Observation
  375. Illustration of Pascal's Theorem
  376. Incenters in Cyclic Quadrilateral
  377. Incidence in Feuerbach's Theorem
  378. Incident Center Circles and Cardioid
  379. Incircles, Circumcircles and Inscribed Angles
  380. Inscriptible and Exscriptible Quadrilaterals
  381. Inscriptible Quadrilateral of Triangle Incenters
  382. Internal Tangents to Three Circles
  383. Intouch Triangle in Poncelet Porism
  384. Invariant Area Sweep Implies Pythagorean Theorem
  385. Inversion: Reflection in a Circle
  386. Inversion in the Incircle
  387. Inversion Tool
  388. Inversion with Negative Power
  389. Invert Two Circles Into Equal Ones
  390. Isogonal and Isotomic Conjugalities
  391. Isogonal Concurrency
  392. Isogonal image of the circumcircle
  393. Isosceles on the Sides of a Triangle
  394. Isosceles Trapezoid In Cyclic Quadrilateral
  395. Isotomy and Isogonality Hand-in-Hand
  396. Iterations in Geometry, an example
  397. Iterations in Geometry, Generalization
  398. Iterations on a Circle Through Three Points
  399. Iterations on Euler Lines
  400. Johnson Circles
  401. Joseph Keech in Bride's Chair
  402. Kiepert's Centroid
  403. Kiepert's Theorem
  404. Kiepert's Triangles Graduate to Ears of Arbitrary Shape
  405. Kürschak's Tile and Theorem
  406. La Hire's Theorem
  407. La Hire's Theorem, a Variant
  408. La Hire's Theorem in Ellipse
  409. Law of Cosines
  410. Law of Cosines: PWW
  411. Law of Cosines: An Unfolded Variant
  412. Law of Cosines by Ancient Sliding
  413. Lighthouse Curves
  414. Line, Circle, and Fixed Points
  415. LCC: Apollonius' Problem with Two Circles and a Line
  416. LLC: Apollonius' Problem with Two Lines and a Circle
  417. Locus of Points in a Given Ratio to Two Points
  418. MacLaurin's Construction of Conics
  419. Malfatti's Problem
  420. Malfatti's Problem, Hart's Solution
  421. Maximum Perimeter Property of the Incircle
  422. MathPro Logo
  423. Maxwell's Theorem
  424. Median in Touching Circles
  425. Medians in a Quadrilateral
  426. Menelaus and Ceva Theorems
  427. Menelaus From Ceva
  428. Midline in Quadrilateral
  429. Midline in Similar Triangles
  430. Midpoint Reciprocity in Napoleon's Configuration
  431. Midpoints and Orthogonality in Isosceles Triangles
  432. Midpoints from Gergonne Triangle
  433. Midpoints of the Lines Joining In- and Excenters
  434. Miguel's Tangents: Modern Day Sangaku
  435. Minkowski's addition of convex shapes
  436. Miquel's Point
  437. Miquel's Point of a 4-line Via Spiral Similarity
  438. Miquel's Theorem for Circles
  439. Mirror on the Wall
  440. Mirror property of the altitudes
  441. Mirror Property of Altitudes via Pascal's Hexagram
  442. Mixtilinear Circles and Concurrence
  443. Monge via Desargues
  444. Monge via Desargues II
  445. More Bottles in a Wine Rack
  446. Morley Constellation
  447. Morley's Equilaterals, Spiridon A. Kuruklis' proof
  448. Morley's Theorem
  449. Morley's Theorem, Cardioids and Such
  450. Morley's Theorem, G. Zsolt Kiss' proof
  451. Morley's Theorem: Taylor and Marr's Proof
  452. Nagel Line
  453. Nagel point
  454. Nagel Point of Medial Triangle
  455. Nagel's Theorem
  456. Napoleon-Barlotti Theorem
  457. Napoleon Theorem by Plane Tessellation
  458. Napoleon's Relatives
  459. Napoleon's Theorem by Transformation
  460. Neuberg Sangaku
  461. Newly Born Pair of Siblings to Archimedes' Twins
  462. Newton's Construction of Conics
  463. Newton's Theorem
  464. Nine Point Circle
  465. No Equilateral Triangles, Please
  466. Nobbs' Points, Gergonne Line
  467. Non-classic Way To Draw Cardioid
  468. No-Pedal Collinearity
  469. Not So Hidden Homotheties
  470. Novel Concurrency of Perpendicular Bisectors
  471. On Bottema's Shoulders
  472. On Bottema's Shoulders II
  473. On Bottema's Shoulders with a Ladder
  474. One More Seven Circles Theorem
  475. One Sided Surface in 4D
  476. Optical Property of Ellipse
  477. Optimal Quadrilateral Inscribed in Square, PWW
  478. Optimization Problem in Acute Angle
  479. Original Taylor and Marr's Proof
  480. Orthogonal Lines, Midpoints, and Collinearity
  481. Orthocenter and Three Equal Circles
  482. Orthocentric System From Rectangle
  483. Orthogonality in Isogonal Conjugacy
  484. Orthogonality in Isosceles Triangles
  485. Orthologic Triangles in a Quadrilateral
  486. Orthopole
  487. Pairs of Homologous Lines under Spiral Similarities
  488. Pairs of Incircles in a Quadrilateral
  489. Pantograph
  490. Pappus' Theorem
  491. Parabola As Envelope of Straight Lines
  492. Parabola As Envelope II
  493. Parabolas Related to the Orthic Triangle
  494. Parabolic Mirror, Illustration
  495. Parabolic Mirror, Theory
  496. Paragon
  497. Parahexagon
  498. Parallel Chords
  499. Parallel Chords in Ellipse
  500. Parallel Lines and Similar Triangles
  501. Parallel Lines in a Cyclic Quadrilateral
  502. Parallel Lines in Rhombus
  503. Parallel Lines in a Triangle
  504. Parallelogram and Similar Triangles
  505. Parallelogram Iterations
  506. Parallelogram Law
  507. Parallelogram Law: A PWW
  508. Parallelogram Law: A Tessellation
  509. Pascal in a Cyclic Quadrilateral
  510. Pascal Lines: Steiner and Kirkman Theorems
  511. Pascal Lines: Steiner and Kirkman Theorems II
  512. Pascal's Theorem
  513. Peaucellier Linkage
  514. Pedal Collinearities
  515. Pedal Parallelogram
  516. Pedal Polygons
  517. Pedal Triangle and Isogonal Conjugacy
  518. Pedoe's Theorem
  519. Pentagon in a Semicircle
  520. Perigal's Proof of the Pythagorean Proposition
  521. Perimeters of Parallelogram And Rhombus
  522. Perpendicular Bisectors in an Inscriptible Quadrilateral
  523. Perpendicular Bisectors in an Inscriptible Quadrilateral II
  524. Pinning Butterfly on Radical Axes
  525. Pivot Theorem
  526. Point common to two similar rectangles
  527. Polar Circle
  528. Pole and Polar with respect to a Triangle
  529. Poles and Polars
  530. Poles, Polars and Orthogonal Circles
  531. Poles, Polars and Quadrilaterals
  532. Polygon from the midpoints of its sides
  533. Polygonal Ears and Triangulation Duals
  534. Polygons Formed by Perpendicular Bisectors
  535. Poncelet's Porism
  536. Poncelet's Porism in Ellipses
  537. Poncelet's Theorem
  538. PCC: Apollonius' Problem with Two Circles and a Point
  539. PLC: Apollonius Problem with a point, a line, and a circle
  540. PPC: Apollonius' Problem with Two Points and a Circle
  541. PPL: An Elementary Solution
  542. PPL: Apollonius' Problem with Two Points and a Line
  543. Problem 4, 1975 USA Math Olympiad and the Radical Axis
  544. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
  545. Problem 4, 1975 USA Math Olympiad: Normals and Tangents
  546. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch
  547. Problem 4 from the IMO 2013 - Synthetic Solution
  548. Problem in an Equilateral Triangle
  549. Problem in an Equilateral Triangle II
  550. Problem In an Isosceles Right Triangle
  551. Problem in Three Squares
  552. Problem of Equal Steps
  553. Problem of Equal Steps II
  554. Problem of polygon construction
  555. Problem with Equilateral Triangles
  556. Product of Rotations
  557. Projections of Convex Quadrilateral
  558. Projections of Isogonal Conjugate are Concyclic
  559. Projections on Internal and External Angle Bisectors
  560. Projective Collinearity in a Quadrilateral
  561. Projective Generalization of Maxwell's Theorem
  562. Projective Proof of Pascal's Theorem
  563. Proof Without Words for the Pythagorean Theorem
  564. Properties of the Circle of Similitude
  565. Property of 6-Parpolygon
  566. Property of Angle Bisectors II
  567. Property of Circumscribed Quadrilaterals
  568. Property of Points Where In- and Excircles Touch a Triangle
  569. Property of Right Trapezoids
  570. Property of the Line IO
  571. Property of the Line IO II
  572. Property of the Line IO: a proof from the Book
  573. Property of Thébault Circles
  574. Proportions with Two Orthocenters
  575. Ptolemy on Hinges
  576. Pythagoras and Vecten Break Japan's Isolation
  577. Pythagoras' from Bottema's
  578. Pythagorean Condition in An Isosceles Right Triangle
  579. Pythagorean Theorem: Subtle Dangers of Visual Proof
  580. Pythagorean Theorem By Euclid
  581. Pythagorean Theorem By Plane Tessellation
  582. Pythagorean Theorem By Hexagonal Tessellation
  583. Pythagorean Theorem By Rearrangement
  584. Pythagorean Theorem By Shearing and Rotations I
  585. Pythagorean Theorem By Shearing and Rotations II
  586. Pythagorean Theorem in an Orthodiagonal Quadrilateral
  587. Quadrilaterals Formed by Perpendicular Bisectors
  588. Quadrilateral with 3 Equal Sides
  589. Quest for Paragon
  590. Radical Axis and Center, an Application
  591. Radical Axis of Circles Inscribed in a Circular Segment
  592. Radical Center
  593. Radius and Construction of a Mixtilinear Circle
  594. Radius of a Circle by Paper Folding
  595. Rectangle in Arbelos
  596. Rectangle Inscribed in Triangle
  597. Rectified, Halved, Sheared, Eyeballs Still Surprise
  598. Reflection in a Triangle Tangent to Parabola
  599. Reflection In Line
  600. Reflections in Ellipse
  601. Reflections of a Line Through the Orthocenter
  602. Reflections of a Point on the Circumcircle
  603. Reflections of the Orthocenter
  604. Regular Polygons in a Triangular Grid
  605. Remarkable Line in a Quadrilateral
  606. Remarkable Line in Cyclic Quadrilateral
  607. Rendezvous
  608. Surprise: Right Angle in Circle
  609. Right Isosceles Triangles on Sides of a Quadrilateral
  610. Rotation Transform
  611. Rotations in Disguise
  612. Rusty Compass Construction of Equilateral Triangle
  613. Salinon: from Archimedes Book of Lemmas
  614. Sanchez's Viviani's Area Analogue
  615. Sangaku Iterations, Is it Wasan?
  616. Sangakus with a Mixtilinear Circle
  617. Sangaku with Three Mixtilinear Circles
  618. Seven Circles Theorem
  619. Squares in Arbelos
  620. Secant, Tangents and Orthogonality
  621. See-Saw Lemma
  622. Segment Trisection
  623. Seven Equal Circles
  624. Shape of Constant Width
  625. Shearing Butterflies in Quadrilaterals
  626. Siamese Triangles
  627. Siamese Triangles II
  628. Similar Triangles on Sides of a Quadrilateral
  629. Similar Triangles on Sides and Diagonals of a Quadrilateral
  630. Simson Line
  631. Simson Line From Isogonal Perspective
  632. Simson Line in Disguise
  633. Simson Lines Of Diametrically Opposite Points
  634. Simsons and 9-Point Circles in Cyclic Quadrilateral
  635. Simultaneous Diameters in Concurrent Circles
  636. sin(N)
  637. Six Circles with Concurrent Pairwise Radical Axes
  638. Six Circles Theorem (Bui Quang Tuan)
  639. Six Circles Theorem (Elkies)
  640. Six Circles Theorem (Evelyn)
  641. Six Concyclic Circumcenters in Fermat's Configuration
  642. Six Concyclic Points
  643. Six Concyclic Points II
  644. Six Concyclic Points via Antipedal Triangle
  645. Six Incircles in an Equilateral Triangle
  646. Six Point Circle
  647. SixPoints, ThreeLines
  648. Soddy Circles and David Eppstein's Centers
  649. Some Properties of Napoleon's Configuration
  650. Spectral Decomposition of a Planar Quadrilateral: an Illustration
  651. Sperner's Lemma
  652. Spira Mirabilis - Wonderful Spiral
  653. Spiral Similarity
  654. Spiral Similarity and CollinearitySpiral Similarity and Collinearity
  655. Square from Four Points, One on Each Side
  656. Square in a Right Triangle
  657. Square Inscribed in a Triangle
  658. Square Inscribed in a Triangle II
  659. Square Inscribed In a Triangle III
  660. Square out of a Quadrilateral
  661. Square, Similarity and Slopes
  662. Squares in Circles
  663. Squares on Sides of a Parallelogram
  664. Squares on Sides of a Quadrilateral
  665. Squares on Sides of a Quadrilateral II
  666. Squinting Eyes Theorem
  667. Star Construction of Shapes of Constant Width
  668. Steiner's Circumellipse in Barycentric Coordinates
  669. Steiner's Chain
  670. Steiner's porism
  671. Straight Edge Construction of Polar
  672. Surprising 60° in Equilateral Triangle
  673. Symmedian and Antiparallel
  674. Symmedian and 2 Antiparallels
  675. Symmedian and the Simson
  676. Symmedian and the Tangents
  677. Symmedian in a Right Triangle
  678. Symmedian in Right Triangle II
  679. Symmedian via Parallel Transversal and Two Circles
  680. Tangency Point of Two Circles
  681. Tangent and Secant
  682. Tangent as a Radical Axis
  683. Tangent Circles and an Isosceles Triangle Tangent Circles and an Isosceles Triangle II
  684. Tangent Circles in a Parallelogram
  685. Tangent in Concurrency
  686. Tangent Lines and Circles in Convex Quadrilateral
  687. Tangents and Diagonals in Cyclic Quadrilateral
  688. Tangents, Perpendiculars and Geometric Mean
  689. Terquem's Theorem
  690. Tesseract
  691. The Midpoint of an Angle Bisector - a Problem by A. A. Zaslavsky
  692. The Plain Butterfly Theorem
  693. The Shortest Crease
  694. Thébault's Problem I
  695. Thébault's Problem II
  696. Thébault's Problem III
  697. Thébault's Problem IV
  698. Thébault's Problem III, Proof
  699. Theorem of Complete Quadrilateral
  700. Theorem of Three Tangents to a Conic
  701. Theorems of Ceva and Menelaus, an Illustrated Generalization
  702. Three Circles through the Incenter
  703. Three Concurrent Circles
  704. Three Congruent Circles by Reflection
  705. Three Congruent Circles by Reflection II
  706. Three Congruent Circles by Reflection III
  707. Three Incircles In a Right Triangle
  708. Three Incircles in a Triangle
  709. Three Isosceles Triangles
  710. Three Orthogonal Circles Through Three Given Points
  711. Three pairs of Parallel Lines and a Conic
  712. Three Parabola Tangents
  713. Three Parabolas with Common Directrix
  714. Three Parallels in a Triangle
  715. Three Points Casey's Theorem
  716. Three Points on a Parabola
  717. Three Roads, Three Travelers
  718. Three Similar Polygons
  719. Three Similar Triangles
  720. Three Similar Triangles (but differently oriented)
  721. Three Similar Triangles (but differently oriented) II
  722. Three Squares and Two Ellipses
  723. Three Tangent Circles
  724. Three Tangents Theorem
  725. Three Tangents Theorem II
  726. Three Tangents, Three Chords in Ellipse
  727. Three Tangents, Three Secants
  728. Toothpick Construction of a Square
  729. Touching Circles
  730. Touching Circles and Concurrency
  731. Touching Circles with Given Cnters II
  732. Transversal Trigonometry
  733. Triangle and Segment Count in a Polygon
  734. Triangle by HM segments
  735. Triangle Construction from Angle, Altitude and Median
  736. Triangle Geometry
  737. Triangle Sliding on Two Straight Lines
  738. Triangles, Squares and Areas from Temple Geometry
  739. Triangles with Equal Area
  740. Triangular Billiard
  741. Trigonometric Form of Ceva's Theorem (a tool for discovery of trigonometric identities and other problems)
  742. Tromino as a Rep-tile
  743. Tucker Circles
  744. Tucker Circles Through Homothety
  745. Twin Segments in Arbelos
  746. Two Butterflies Theorem
  747. Two Butterflies Theorem II
  748. Two Butterflies Theorem III
  749. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  750. Two Circles and One More
  751. Two Circles and Two More
  752. Two Circles in a Parallelogram
  753. Two Circles Inscribed in a Parallelogram
  754. Two Circles in a Square
  755. Two Circles in a Square II
  756. Two Circles in an Angle
  757. Two Circles on Angle Bisector
  758. Two Circles on a Side of a Triangle
  759. Two Circles, Tangents and Two Collinearities
  760. Two Circumcircles And Two Pairs of Parallels
  761. Two Circumcircles in Triangle
  762. Two Congruent Circles by Reflection
  763. Two Conics, Pascal, Chasles, and Cross-Ratio
  764. Two Homotheties in a Parallelogram
  765. Two Lines - Two Circles
  766. Two Pairs of Parallel Lines in a Triangle
  767. Two Pencils of Parallel Lines
  768. Two Proofs of MenelausTheorem
  769. Two Quadrilaterals
  770. Two Quadruplets of Concyclic Points
  771. Two Quadruplets of Tangent Circles
  772. Two Rectangles in a Rectangle
  773. Two Similar Triangles with Parallel Sides
  774. Two Simsons in a Triangle
  775. Two Squares and Another Square
  776. Two Triples of Similar Triangles
  777. Two Tangents to Parabola
  778. Two Triangles Inscribed in a Conic
  779. Two Triangles Inscribed in a Conic - with Elementary Solution
  780. Two Triangles Inscribed in a Conic - with Solution
  781. Two Triangles With Common Base and Altitude
  782. Two Triples of Concurrent Cevians and a Conic
  783. Two Triples of Concurrent Circles
  784. Van Schooten's Locus Problem
  785. Varignon Parallelogram
  786. Vecten-Nikolin Hexagon
  787. Vecten's Collinearity
  788. Vecten's Mesh
  789. Vecten's Theorem
  790. Versatile Theorem
  791. Vietnamese Lemma towards an Extension of a Japanese Theorem
  792. Weitzenböck Inequality
  793. What does one see inside a spherical mirror?
  794. When a Triangle is Equilateral
  795. William Wallace's 1803 Statement of the Butterfly Theorem
  796. Wittenbauer's Parallelogram
  797. Y. Sawayama's Lemma
  798. Yet Another Concurrence on the 9-Point Circle
  799. Yet Another Seven Circles Theorem
  800. Zaslavsky's theorem

    Miscellaneous Demonstrations

  801. Calculation of the Digits of pi by the Spigot Algorithm
  802. Calculation of the Digits of pi (Faster Version)
  803. Convex Polygon Is the Intersection of Half Planes
  804. Disapperances à la Sam Loyd
  805. Disappearing Lines
  806. Disappearing Rectangles
  807. Distance Between Strings
  808. Experimentation with Dynamic Geometry Software: An Example
  809. Genetic Algorithm Solves the Toads and Frogs Puzzle
  810. Impossible Fork
  811. Impossible Frame
  812. Morphing
  813. Order and Chaos in Multiple Pendulums
  814. Projectile Trajectory
  815. Random Dot Stereograms
  816. Ratchet Effect
  817. Rotating Gears
  818. Sorting Algorithms
  819. Structural Constellation
  820. Watt's and Chebyshev's Linkage

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Copyright © 1996-2018 Alexander Bogomolny

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