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. Square in a Circular Sector
  124. Proizvolov's Identity
  125. Pythagorean Triples
  126. Pythagorean Triples Calculator
  127. Quasigroup on a Triangular Grid
  128. Rectangle on a Chessboard
  129. Right Replacement
  130. Self-descriptive Strings
  131. Self-documenting Sentences
  132. Sierpinski's Gasket and Dihedral Symmetry
  133. Signs and Sums in a Sequence
  134. Signs in a Matrix
  135. Simple Cellular Automaton
  136. Simple Somos Sequence Calculator
  137. Simpson Paradox
  138. Sine of a Sum Formula
  139. Solitaire in Two Colors
  140. Squares and Circles (parity, invariance)
  141. Squares, Circles, and Triangles (modular arithmetic, invariance)
  142. Steinhaus' Problem
  143. Sum of Consecutive Integers is Triangular
  144. Sum of Consecutive Odd Numbers is Square
  145. Sum of Consecutive Triangular Numbers is Square
  146. Sums of Geometric Series - Proofs Without Words
  147. Swapping Rows and Columns
  148. Toads And Frogs Puzzle (counting, logic, problem solving)
  149. Two Colors in Two Rows
  150. Two Colors in Two Rows II
  151. Voronoi Diagram
  152. What's next?
  153. Word Problems


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


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


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

    Visual Illusions

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


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

    Math Magic

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


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

    Combinatorial Games

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

    Puzzles & Games

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

    Social Science

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

    Elementary Geometry

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


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

    Miscellaneous Demonstrations

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

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Copyright © 1996-2018 Alexander Bogomolny


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