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


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

    Miscellaneous Demonstrations

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

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Copyright © 1996-2017 Alexander Bogomolny


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