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 Straightedge Construction of the Midpoint of a Chord Common to Two Circles
  465. A Theorem of M. Mansion
  466. A Three Pegs Question
  467. A Triangle With a 45 Degrees Angle in Square
  468. Acute Angle Bisectors in a Right Triangle
  469. Al-Nayrizi's Construction
  470. An Appearance of a 30-60-90 Triangle
  471. An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem
  472. An Extra Triple of Equilateral Triangles for Napoleon
  473. An Unexpected Rhombus
  474. Angle: An Illustrated Classification
  475. Angle Bisector
  476. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise
  477. Angle Bisectors in Ellipse
  478. Angle Bisectors in Ellipse II
  479. Angle Bisectors On Circumcircle
  480. Angle Subtended by a Diameter
  481. Angles Inscribed in an Absent Circle
  482. Anticenter and Orthocenters
  483. Antiparallel
  484. Antiparallel via Three Reflections
  485. Archimedes' Law of the Lever
  486. Area of Median Triangle I
  487. Area of Median Triangle II
  488. Area of Parallelogram
  489. Area of the Intersection of Two Equal Rectangles
  490. Area of the Union of Two Squares
  491. Area of Triangle
  492. Areas between Two Parallelograms
  493. Areas in Square by Dissection
  494. Areas in Two Parallelograms and a Translation
  495. Arithmetic in Disguise
  496. Bespectacled Eyeballs Extension
  497. Between Major and Minor Circles
  498. Bottema with a Product of Rotations
  499. Bottema with Similar Triangles
  500. Bottema's Theorem - Proof Without Words
  501. Building a Bridge
  502. Building Bridges
  503. Carpet With a Hole
  504. Carpets in Triangle
  505. Center of Gravity and Motion of Material Points
  506. Chain of Four Intersecting Circles
  507. Characterization of Trapezoid
  508. Circle and Hyperbola as Lighthouse Curves II
  509. Circle Chains on Napoleon Triangles
  510. Circle Rotations and Fixed Points
  511. Circles and Semicircles in Rectangle
  512. Circle in a Square Inscribed in a Circle
  513. Circles with Equal Collinear Chords
  514. Circumcenter and Orthocenter
  515. Circumcenter on Angle Bisector
  516. Classification of Quadrilaterals
  517. Common Chord and a Tangent
  518. Concurrence in Arbelos
  519. Concurrence of Ten Nine-Point Circles
  520. Concurrence on a Circle
  521. Concurrent and Parallel Lines in Parallelogram
  522. Concyclicity from Collinearity
  523. Concyclicity in Rectangle
  524. Compass Only Construction - A Chord Tangent to an Inner circle
  525. Concyclic Points from Midpoint of an Arc
  526. Concyclicity in Two Semicircles
  527. Conic from Parallel Chords
  528. Conic in Hexagon
  529. Conic Sections as Loci of Points
  530. Conics in Triangle
  531. Construct Triangle by Angle, Altitude and Median
  532. Construct Triangle by Angle, Median, and Circumradius
  533. Construction: Parallel Line through Point
  534. Construction of a Right Triangle
  535. Construction of an Oval - the Roman Way
  536. Construction of the Angle Bisector
  537. Construction of the Perpendicular Bisector
  538. Cross Points in a Polygon
  539. Cutting a Hole in Half
  540. Cyclic Quadrilateral, Angle Bisector And Isosceles Triangle
  541. Dancing Rectangles Model Auxetic Behavior
  542. Dancing Squares or a Hinged Plane Tessellation
  543. A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
  544. Diagonal Count
  545. Dissection of a Vase
  546. Dissection of Cyclic Quadrilateral
  547. Divide a Circle into N Parts of Equal Area
  548. E. W. Dijkstra's Proof of the Pythagorean Theorem
  549. Eight Equal Tangents
  550. Equal Areas in Equilateral Triangle
  551. Equal And Perpendicular Segments in a Square
  552. Equal Areas in Regular 2n-gons
  553. Equal Chords in Crossing Circles
  554. Equal Circles, Medial Triangle and Orthocenter
  555. Equidecomposition of a Rectangle and a Square
  556. Equidecomposition of Two Rectangles
  557. Equidecomposition of a Triangle and a Rectangle
  558. Equidecomposition of a Triangle and a Rectangle II
  559. Equidecomposition of Two Parallelograms
  560. Equilateral Triangle on a Closed Curve
  561. Equilateral Triangle on Parallel Lines
  562. Equilateral Triangle on Parallel Lines II
  563. Equilateral Triangle on Three Lines
  564. Equilateral Triangles formed by Circumcenters
  565. Equilateral Triangle in Equilateral Triangle
  566. Equilic Quadrilateral I
  567. Equilic Quadrilateral I, A Variation
  568. Equilic Quadrilateral II
  569. Euclid's Segment Division
  570. Existence of the Circumcenter
  571. Existence of the Euler Line: An Elementary Proof
  572. Existence of the Orthocenter
  573. Extras in Bottema's Configuration
  574. Eye-to-Eye Theorem I
  575. Eye-to-Eye Theorem II
  576. Fibonacci Bamboozlement
  577. Famous Theorem
  578. Finding Circle Center By Ruler Alone
  579. Five Squares in Complex Numbers
  580. Fixed Point in Isosceles and Equilateral Triangles
  581. Fixed Point on Perpendicular Bisector
  582. Focal Definition of Ellipse
  583. Focus and Directrix of Ellipse
  584. Folding Square in a Line through the Center
  585. Four Circles and a Line
  586. Four Hinged Squares
  587. From Arbitrary Hexagon to Regular One
  588. From Perpendicular Center Lines to Concyclic Points
  589. From Straight Line to Regular Hexagon
  590. From Arbitrary Pentagon to Regular One
  591. Garcia's Two Circles Lemmas
  592. Geoboard
  593. Half Turn, Reflection in Point
  594. Harmonic Mean Sangaku
  595. Hinged Greek Cross Tessellation
  596. Hippocrates' Squaring of a Lune
  597. Homothety in Three Tangent Circles
  598. Homothety in Three Tangent Circles II
  599. How to Construct Tangents from a Point to a Circle
  600. Hunting Right Angles
  601. Huygens' proof of the Pythagorean Theorem
  602. In the Spirit of Thebault I
  603. Inscribed and Central Angles in a Circle
  604. Inscribed Angles
  605. Inscriptible Quadrilateral: An Illustration
  606. Integer Triangles with Two Angles in the Ratio 1 : 2
  607. Intersecting Chords Theorem - a Visual Proof
  608. Intersecting Chords Theorem - Hubert Shutrick's PWW
  609. Intersections of a Circle with the Four Quadrants
  610. Invariance in Orthodiagonal Quadrilaterals
  611. Is X a Midpoint of a Chord
  612. Isogonal Lemma
  613. Isogonal Concurrencies
  614. Isoperimetric Theorem For Quadrilaterals
  615. Isoperimetric Theorem For Quadrilaterals II
  616. Isosceles Configuration in Triangle
  617. Join Circles by Given Segment
  618. Joined Common Chords of Napoleon's Circumcircles
  619. Law of Cosines - Another PWW
  620. Lean Napoleon's Triangles
  621. Line Not Through a Center of Similarity
  622. Line Through a Center of Similarity
  623. Looking Back at Bottema
  624. Maimed Cake
  625. Mathematics in Pizzeria
  626. Mickey Might Be a Red Herring in the Mickey Mouse Theorem
  627. Midline in Trapezoid
  628. Midline in Triangle
  629. Miguel's Area of Square
  630. Miguel Ochoa's Chords And Tangents
  631. Miguel Ochoa's van Schooten is a Slanted Viviani
  632. Miguel Ochoa's van Schooten Like Theorem
  633. Miguel Ochoa's van Schooten Like Theorem II
  634. Miguel Ochoa's van Schooten Like Theorem III
  635. Miquel Circumcenters
  636. Napoleon's Hexagon
  637. Napoleon's Theorem: A third proof with complex numbers
  638. Napoleon's Theorem via Two Rotations
  639. Napoleon on Hinges
  640. Napoleon on Hinges in GeoGebra
  641. Neuberg Cubic
  642. Nine Point Circle: an Elementary Proof
  643. Octagon in Parallelogram
  644. Octagon in Parallelogram: Overlapping Regions
  645. Olof Hanner's Jigsaw Puzzle
  646. One More Property of Equilateral Triangles
  647. Orthic Semiperimeter
  648. Orthocenter and Circumcenter Are Isogonal Conjugate
  649. Orthocenters of Two Triangles Sharing Circumcenter and Base
  650. Orthodiagonal and Cyclic Quadrilaterals
  651. Orthogonality in Two Squares
  652. Pappus' Generalization of Euclid I.47
  653. Parabolic Reciprocity
  654. Parallel Chords II
  655. Parallel Chords in Crossing Circles
  656. Parallel Lines in a Quadrilateral
  657. Parallel Lines in a Quadrilateral II
  658. Parallelogram and Four Equilateral Triangles
  659. Parallelograms among Quadrilaterals
  660. Parallelogram and Ellipses
  661. Parallelogram in Four Circles
  662. Parallelogram in Parallelogram And in Projection
  663. Parallelogram with Side Lines through Fixed Points
  664. Pascal: Necessary and Sufficient
  665. Pedal Similarity
  666. Pedal Triangles of Inverse Points
  667. Perimeters of Convex Polygons, One within the Other
  668. Playing on Flanks
  669. PLL: Apollonius' Problem with Two Lines and a Point
  670. Polygon Metamorphosis
  671. Pompeiu's Theorem
  672. Praying Eyes Theorem
  673. Problem 1 from the EGMO2017
  674. Proclivity of Conjoint Archimedean Twins for Proliferation
  675. Properties of Flank Triangles - Second Proof by Symmetric Rotation
  676. Properties of Flank Triangles - Proof with an Asymetric Rotation
  677. Property of Angle Bisectors
  678. Property of Angle Bisectors III
  679. Property of Internal Angle Bisector - Hubert Shutrick's PWW
  680. Property of Rhombi
  681. Property of Right Isosceles Triangle
  682. Property of Semicircles
  683. Pure Angle Chasing
  684. Pure Angle Chasing II
  685. Pure Angle Chasing III
  686. Pythagorean Theorem By Dissection
  687. Pythagorean Theorem By Dissection II
  688. Pythagorean Theorem by Hinged Dissection
  689. Pythagorean Theorem by Hinged Dissection I
  690. Pythagorean Theorem by Hinged Dissection II
  691. Pythagorean Theorem: J. Adams' Proof by Dissection
  692. Pythagorean Theorem: Versluys' Proof by Dissection
  693. Pythagoras' Theorem By Sheer Shearing
  694. Pythagoras' Theorem By Similarity
  695. Pythagoras' Theorem Solely By Sliding
  696. Pythagorean Theorem via an Isosceles Triangle
  697. Pythagorean Segment Division
  698. Quadrilateral With Equal Opposite Sides And Angles
  699. Rectangles among Parallelograms
  700. Rectangular Areas in Circle
  701. Reflection in the Circumcenter
  702. Regular Hexagon and Triangle Joined at Vertex
  703. Reim's Similar Coins I
  704. Reim's Similar Coins II
  705. Reim's Similar Coins III
  706. Reim's Similar Coins IV
  707. Rhombus in a Cyclic Quadrilateral
  708. Right Triangles on Sides of a Square
  709. Rhombus in Circles
  710. Sangaku with Angle between a Tangent and a Chord
  711. Secant Angles
  712. Secant Angles II
  713. Secant, Normal, Tangent
  714. Segment Trisection II
  715. Segment Trisection Induced by Parallels to Medians
  716. Semi-regular Tessellation on Hinges A
  717. Semi-regular Tessellation on Hinges B
  718. Semi-regular Tessellation on Hinges C
  719. Seven Concyclic Points
  720. Shearing a Polygon into a Triangle of Equal Area
  721. Shifted by the Inradius Tangent Collinearities
  722. Similar Triangles in Crossing Circles
  723. Simple Property of Circle Through the Incenter
  724. Simple Quadrilaterals Tessellate the Plane
  725. Shearing and Translation in Pythagorean Pants
  726. Some Properties of Common Tangents
  727. Spiral Similarity Leads to Equilateral Triangle
  728. Square, 45 Degrees Angle and Pythagoras' Look-alike
  729. Square, Semicircle, and Other Curves
  730. Square From Nowhere
  731. Square in a Circle Inscribed in a Square
  732. Square in Chair
  733. Squaring a Rectangle
  734. Sum of Two Squares
  735. Symmetries in Triangle II
  736. Symmetry in an Asymmetric Configuration
  737. Tangent Circles, Similarity and Equal Segments
  738. Tangent Curiosity in Equilateral Triangle
  739. Thales' Theorem
  740. Thales on Angle Bisectors
  741. The Eutrigon Theorem
  742. The Shape of the Colosseum
  743. Three Collinear and Four Concyclic Points
  744. Three Common Chords in Three Concurrent Circles
  745. Three lines at 60 degres and Circumcenters
  746. Three Touching Circles
  747. Translated Triangles
  748. Translation Transform
  749. Triangle Areas in a Parallelogram
  750. Triangle Areas in a Parallelogram II
  751. Triangle Classification
  752. Trigonometric Functions
  753. Triangular Areas in Circle
  754. Twin Segments in Four Semicircles
  755. Two Altitudes, One Midpoint
  756. Two Color Coloring of the Plane
  757. Two Common Chords in Three Concurrent Circles
  758. Two Diameters and Longest Common Chord
  759. Two Equilateral Triangles
  760. Two Intersecting Circles
  761. Two Parallels in a Triangle and One More
  762. Two Pascals Merge into One
  763. Two Properties of Flank Triangles - First Proof by Symmetric Rotation
  764. Two Properties of Flank Triangles - A Proof with Complex Numbers
  765. Two Properties of Flank Triangles - and a Third One
  766. Two Regular Pentagons Joined at Vertex
  767. Two Symmetric Triangles Are Directly Equidecomposable
  768. Two Symmetric Triangles Are Directly Equidecomposable II
  769. Two Symmetric Triangles Are Directly Equidecomposable III
  770. Two Symmetric Triangles Are Directly Equidecomposable IV
  771. Two Touching Circles
  772. Van Khea's Quickie
  773. Van Schooten's and Pompeiu's Theorems
  774. Viviani's Theorem
  775. Viviani's Theorem II
  776. Viviani in Isosceles Triangle
  777. Yet Another Generalization of Bottema's Theorem


  778. Angles in a Cube I
  779. Angles in a Cube II
  780. Angles in a Cube III
  781. Bimedians in a Regular Tetrahedron
  782. Cut the Cone
  783. Cut the Cube
  784. Cut the Cylinder
  785. Knots on a Torus
  786. Möbius' Strip
  787. Multifaceted Cork
  788. One Sheet Hyperboloid
  789. Platonic Solids - Regular Polyhedra
  790. Other Solids

    Outline (Fill-It-In) Mathematics

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

    Geometric Fallacies

  792. A Circle With Two Centers
  793. All Triangles Are Isosceles
  794. Curry's Paradox
  795. Delian Problem Solved
  796. A Faulty Dissection
  797. Hooper's Paradox
  798. Is Every Trapezoid Parallelogram?
  799. John Sharp's Paradox
  800. Langman's Paradox
  801. Popping A Square
  802. Rouse Ball's Fallacy
  803. Sam Loyd's Son's Dissection
  804. Two Perpendiculars From a Point to a Line



  1. 120° Breeds 90°
  2. 2N-Wing Butterfly Theorem
  3. 3 Isosceles Trapezoids
  4. 5-Star and A Circle
  5. 6 to 9 Point Circle
  6. 60° Breeds 90°
  7. 9-point Circle as a Locus of Concurrency
  8. 9 Point Center on Angle Bisector
  9. A Chain of Touching Circles in a Polygon (à la Quang Tuan)
  10. A Chain of Touching Circles in a Polygon (à la Evelyn)
  11. A Circle Related to Incenter and Circumcenter
  12. A Circle Rolling in an Equilateral Triangle
  13. A Conic in a Triangle
  14. A Generalization of Simson Line
  15. A Geometry Problem from 2012 IMO - Problem 5
  16. A Line in Triangle Through the Circumcenter
  17. A Parallelogram in Triangle
  18. A Problem in Pentagon with Right Angles
  19. A Problem in Pentagon with Supplementary Angles
  20. A Problem in Three Pentagons
  21. A Property of Angle Bisectors
  22. A Sangaku Follow-Up on an Archimedes' Lemma
  23. A Sangaku: Two Unrelated Circles
  24. A Sangaku with an Egyptian Attachment
  25. A Sangaku with Many Circles and Some
  26. A 60° Rhombus in a 60° Isosceles Trapezoid
  27. A Problem in a Configuration of Three Squares
  28. A Stronger Triangle Inequality
  29. A Triangle in a Rhombus with a 60 Degrees Angle
  30. A Triangle of Antreas Hatzipolakis
  31. Abutting Equilateral Triangles on a Line I
  32. Adams' Circle
  33. Addition of Radius-Vectors
  34. Altitudes and the Power of a Point
  35. An Equilateral Triangle In Parallel Lines
  36. An Euclidean Construction with Inversion
  37. An Isoperimetric Theorem
  38. An Unexpected Pair of Similar Triangles Which Are Equal
  39. Angle Bisector in Equilateral Trapezoid
  40. Angle Bisector in Parallelogram
  41. Angle Bisector in Touching Circles
  42. Angle Bisectors and Perpendiculars in a Quadrilateral
  43. Angle Bisectors in a Quadrilateral
  44. Angle Preservation Property
  45. Angle Trisection by Archimedes
  46. Angle Trisection by Hippocrates
  47. Angle Trisection by Paul Vjecsner
  48. Angle Trisectors on Circumcircle
  49. Angles in Triangle Add to 180°
  50. Another Concurrence on the 9-Point Circle
  51. Another Geometry Problem from 2012 IMO - Problem 1
  52. Another Equilateral Triangle in Napoleon's Configuration
  53. Another Pair of Twins in an Arbelos
  54. Another Property of the 9-Point Circle
  55. Another Seven Circles Theorem
  56. Anticomplementary Triangle Surprise
  57. Antiparallel and Circumradius
  58. Apollonian Circle for Two Lines and a Circle
  59. Apollonian Circles Theorem
  60. Apollonius Problem
  61. Application of Ceva's Theorem
  62. Archimedean Siblings out of Wedlock, i.e., Arbelos
  63. Archimedes' Quadruplets
  64. Archimedes Triangle and Squaring of Parabola
  65. Archimedes' Twin Circles and a Brother
  66. Area of an Arbelos
  67. Areas and Centroid in a Triangle
  68. Areas in Three Squares
  69. Areas in Triangle
  70. Areas in Triangle II
  71. Arithmetic Mean Sangaku
  72. Arithmetic Mean Sangaku II
  73. Around the Incircle
  74. Around the Incircle in a Polygon
  75. Asian Pacific Kite
  76. Asymmetric Propeller I
  77. Asymmetric Propeller II
  78. Asymmetric Propeller III
  79. Asymmetric Propeller by Plane Tiling
  80. Barycentric Coordinates: a Tool
  81. Batman's Problem
  82. Bevan's Point and Theorem
  83. Bisectal Circle
  84. BMO 2014, Problem 3
  85. Book of Lemmas: Proposition 1
  86. Book of Lemmas: Proposition 2
  87. Book of Lemmas: Proposition 3
  88. Book of Lemmas: Proposition 5
  89. Book of Lemmas: Proposition 6
  90. Book of Lemmas: Proposition 9
  91. Book of Lemmas: Proposition 10
  92. Book of Lemmas: Proposition 12
  93. Book of Lemmas: Proposition 13
  94. Böttcher's Proof of the Pythagorean Theorem
  95. Böttcher's Proof Variant
  96. Bottema Shatters Japan's Seclusion
  97. Bottema's Theorem
  98. Bottles in a Slanted Rack
  99. Bottles in a Wine Rack
  100. Bottles in a Wine Rack: Proofs and Generalizations
  101. Brahmagupta's Theorem
  102. Brianchon in Ellipse
  103. Brianchon's Theorem
  104. Bride's Chair
  105. Broken Chord Theorem
  106. Broken Chord Theorem by Paper Folding
  107. Broken Chord Theorem: G. Patruno's proof
  108. Broken Chord Theorem: S. Anderson's proof
  109. Butterflies in a Pencil of Conics
  110. Butterfly in Ellipse
  111. Butterfly in Parabola
  112. Butterfly Problem
  113. Cabart's Collinearity
  114. Cantor's Theorem
  115. Cardioid As Envelope of Circles
  116. Cardioid As Epicycloid
  117. Carnot's Theorem
  118. Carpets in a Parallelogram
  119. Carpets in a Quadrilateral
  120. Carpets in a Quadrilateral II
  121. Carpets Theorem
  122. Carpets Theorem With Parallelograms
  123. Casey's Theorem
  124. Cassini's Ovals and Geometric Optimization
  125. CCC: Apollonius' Problem with Three Intersecting Circles
  126. CCC: Gergonnes's Solution to Apollonius' Problem
  127. Center Circles and Their Chains
  128. Centroids in a Polygon
  129. Ceva and Menelaus Theorems for Angle Bisectors
  130. Ceva's Theorem
  131. Ceva's Theorem (PWW)
  132. Cevian Cradle
  133. Cevian Cradle II
  134. Cevian Nest
  135. Cevian Parallelogram
  136. Cevian Triangle
  137. Chain of Circles on a Chord
  138. Chain of Inscribed Circles
  139. Chain of Six Intersecting Circles
  140. Chasing Angles in Pascal's Hexagon
  141. Chasles' Theorem, a Proof
  142. Cherchez le quadrilatère cyclique
  143. Cherchez le quadrilatère cyclique II
  144. Chords, Concurrency and Orthic Triangle
  145. Chords in Cardioid
  146. Circle and Hyperbola as Lighthouse Curves
  147. Circle Centers on Radical Axes
  148. Circle Chains and Inscribed Angles
  149. Circle Concurrence on Circumcircle/a>
  150. Circle Concurrency and Spiral Similarity
  151. Circle Crosses Triangle
  152. Circle Division by Euclid and Leonardo
  153. Circle Inscribed in a Circular Segment
  154. Circle of Similitude
  155. Circle through the Circumcenter
  156. Circle through the Incenter
  157. Circle through the Incenter And Antiparallels
  158. Circle-Stacking Theorem
  159. Circle, Isosceles Triangle and Fixed Point
  160. Circles And Parallels
  161. Circles and Reflections in a Triangle
  162. Circles Cover a Quadrilateral
  163. Circles in a Regular Polygon
  164. Circles in Morley's Triangles
  165. Circles Lined on the Legs of a Right Triangle
  166. Circles On Cevians
  167. Circles Tangent to Circumcircle
  168. Circles Tangent to Medians at the Centroid
  169. Circles through the Orthocenter
  170. Circles with Tangent Diameters
  171. Circumcevian Triangle
  172. Cleavance Center
  173. Cleaver
  174. Clifford Chain
  175. Clifford's Lemma
  176. Coaxal Circles on Perpendicular Bisector
  177. Coaxal Circles Theorem
  178. Colburn's Proof of the Pythagorean Theorem
  179. Collinear Intersections and Products of Ratios
  180. Collinearity from the IMO 2013 (Problem 4)
  181. Collinearity via Concyclicity
  182. Collinearity with the Orthocenter
  183. Common Centroids Lead to Equilateral Triangle
  184. Common Tangents to Two Circles I
  185. Common Tangents to Two Circles II
  186. Complete Quadrilateral
  187. Concurrence in Equilateral Triangle
  188. Concurrence in Right Triangle
  189. Concurrence Not from School Geometry
  190. Concurrence of the Altitudes As Seen from 3D
  191. Concurrency in Analytic Geometry
  192. Concurrency in Arbelos
  193. Concurrency in the Intouch Triangle
  194. Concurrent Cevians and a Conic through Their Feet
  195. Concyclic Circumcenters: A Dynamic View
  196. Concyclic Points in a Triangle
  197. Concyclic Incenters in Bicentric Quadrilateral
  198. Concyclic Points at the IMO 2014
  199. Concyclic Points in Bride's Chair
  200. Concyclic Points in Equilateral Bumps
  201. Concyclic Points in Inscriptible Quadrilateral
  202. Concyclic Points of Two Ellipses with Orthogonal Axes
  203. Conic in Mixtilinear Incircles
  204. Conjugate Diameters in Ellipse
  205. Construct Triangle by Angle Bisector, Altitude, and Side
  206. Construction and Properties of Mixtilinear Incircles
  207. Construction and Properties of Mixtilinear Incircles 2
  208. Construction of Conics from Pascal's Theorem
  209. Construction of Thebault Circles
  210. Consecutive Isosceles Decomposition
  211. Conway's Circle
  212. Copernicus' Theorem
  213. Crossed-Lines Construction of Shapes of Constant Width
  214. Cutting Triangle in Two by a Line through a Point
  215. Cyclic Hexagon
  216. Cyclic Incenters
  217. Cyclic Quadrilateral, Concurrent Circles and Collinear Points
  218. Cycloids
  219. Dao's Archimedean Twins
  220. Dao's Variant of Thébault's First Problem
  221. David Richeson's Extention of an Old Japanses Theorem
  222. Diameters and Chords
  223. Directly Similar Figures
  224. Desargues' Hexagon
  225. Desargues' Theorem
  226. Discovery of Duane DeTemple
  227. Dissection of Triangle into Rhombus
  228. Dissection of Triangle into Rhombus by Hubert Shutrick
  229. Divide Triangle by Lines Parallel to Base
  230. Dividing a Segment into N parts: Besteman's Construction
  231. Dividing a Segment into N parts: Besteman's Construction II
  232. Dividing a Segment into N parts: GLaD Construction
  233. Dividing a Segment into N parts: SaRD Construction
  234. Dividing Evenly a Quadrilateral
  235. Dividing Evenly a Quadrilateral II
  236. Double Generation Theorem
  237. Douglas' Theorem (complex variables, linear algebra)
  238. Droz-Farny Circles
  239. Droz-Farny Line Theorem
  240. Droz-Farny Theorem - an Analytic Solution
  241. Dual of Maxwell's Theorem
  242. Dual to Pappus' Theorem
  243. Dynamic Construction of Ellipse and Other Curves
  244. Dynamic Software as Serendipity Enhancement
  245. Easy Construction of Bicentric Quadrilateral
  246. Easy Construction of Bicentric Quadrilateral II
  247. Eight Point Circle
  248. Ellipse, how to draw
  249. Ellipse Between Two Circles
  250. Ellipse in Arbelos
  251. Ellipse Touching Sides of Triangle at Midpoints
  252. Emil Stoyanov's New Year's Problem
  253. Equal Angles in Two Circles
  254. Equal Angles in Two Circles II
  255. Equal Areas In Parallelogram à la Pythagoras
  256. Equal Incircles Theorem
  257. Equichordal Curves
  258. Equilateral and 3-4-5 Triangles
  259. Equilateral Triangle, Straight Line and Tangent Circles
  260. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  261. Equilateral Triangles on Segments of Equilic Quadrilateral
  262. Equilateral Triangles On Sides of a Parallelogram
  263. Equilateral Triangles On Sides of a Parallelogram II
  264. Equilateral Triangles on Sides of a Parallelogram III
  265. Equilateral Triangles on Sides of a Quadrilateral
  266. Escher's Theorem
  267. Euler Line Cuts Off Equilateral Triangle
  268. Euclid I.43
  269. Euclid I.43 Extended
  270. Euclidean Construction of Center of Ellipse
  271. Euclidean Construction of Tangent to Ellipse
  272. Excircles in Trapezoid
  273. Exeter Point
  274. Extouch Triangle in Poncelet Porism
  275. Extra Feature of Van Aubel Configuration
  276. Extremal Distance Ratio
  277. Eyeball Theorem
  278. Eyeballing a Ball
  279. Eyeballs Projected
  280. External Angle Bisector
  281. Fagnano's problem
  282. Fagnano's problem II
  283. Fagnano's Problem, Third Solution
  284. Fagnano's Problem, Fourth Solution
  285. Fagnano's Problem in Reverse
  286. Fermat's Hexagon
  287. Fermat point and 9-point Centers
  288. Fermat's Point
  289. Fermat Points and Concurrent Euler Lines I
  290. Fermat Points and Concurrent Euler Lines II
  291. Feuerbach's theorem
  292. Feuerbach's Theorem: A Proof
  293. Final Chapter of the Asymmetric Propeller Story
  294. Find a Common Chord of Given Length
  295. Find the Center of a Given Circle
  296. First Vecten Point
  297. Five Cardioids
  298. Five Concyclic Points
  299. Five Incircles in a Square
  300. Five Incircles Theorem
  301. Fixed Point of Circles Orthogonal to the Given One
  302. Fixed point on a rod rotating around a triangle
  303. Focal Properties of Parabola
  304. Focus on the Eyeball Theorem
  305. Foot of Altitude and Minimum Distance
  306. For Equality Choose Angle Bisector
  307. Four 9-Point Circles in a Quadrilateral
  308. Four Centroids and Parallels
  309. Four Circles In a Triangle
  310. Four Concurrent Lines in a Cyclic Quadrilateral
  311. Four Crossing Circles
  312. Four Incident Circles
  313. Four Incircles in Equilateral Triangle
  314. Four Isosceles Triangles Hinged at Vertices
  315. Four Pedal Circles
  316. Four Touching Circles
  317. Four Touching Circles II
  318. Four Touching Circles III
  319. The Four Touching Circles Problem IV
  320. Four Triangles, One Circle
  321. Four Turtles (or Bugs, or Dogs, etc.)
  322. François Viète's Reduction of CCC to PCC
  323. Friendly Kiepert's Perspectors
  324. From Foci to a Tangent in Ellipse
  325. Fundamental Theorem of 3-Bar Motion
  326. Further properties of Van Aubel Configuration
  327. Fuss' Theorem
  328. Garcia's Archimedean Quadruplets
  329. Garcia-Feuerbach Collinearity
  330. Generalizing a Romanian Olympiad Problem
  331. Geometric/Trigonometric Observation/Exercise
  332. Geometric Optimization from the Asian Pacific Mathematical Olympiad
  333. Gergonne and Medial Triangles Are Orthologic
  334. Gergonne and Soddy Lines Are Perpendicular
  335. Gergonne in Ellipse
  336. Glide Reflection
  337. Golden Ratio in an Irregular Pentagon, Construction I
  338. Golden Ratio in an Irregular Pentagon, Construction II
  339. Golomb's inductive proof of a tromino theorem
  340. Gothic Arc
  341. Grebe: from Ladies' Diary to Carroll's Pillow
  342. Grebe's Theorem
  343. Greg Markowsky's Problem for Parabola
  344. Griffiths Points in Cyclic Quadrilateral
  345. Harmonic Ratio
  346. Hart Circle
  347. Hart's Inversor
  348. Haruki's Lemma
  349. Heron's Formula: a Proof
  350. Heron's Problem
  351. Heron in Trapezoid
  352. Hexagon Parallel to Medial Triangle
  353. Hexagon Parallel to Orthic Triangle
  354. Hidden Orthogonality
  355. Hinged Squares
  356. Hinged Squares II
  357. Hjelmslev Theorem
  358. Hjelmslev's Theorem on Canvas
  359. Hofstadter Triangles and Points
  360. Homologous Lines under Three Spiral Similarities
  361. Homothety
  362. Homothety between In- and Excircles
  363. Homothety in Equilateral Triangles
  364. How to Construct a Radical Axis
  365. How to Construct Orthogonal Circle
  366. Hypocycloid Families
  367. Ian McGee's Observation
  368. Illustration of Pascal's Theorem
  369. Incenters in Cyclic Quadrilateral
  370. Incidence in Feuerbach's Theorem
  371. Incident Center Circles and Cardioid
  372. Incircles, Circumcircles and Inscribed Angles
  373. Inscriptible and Exscriptible Quadrilaterals
  374. Inscriptible Quadrilateral of Triangle Incenters
  375. Internal Tangents to Three Circles
  376. Intouch Triangle in Poncelet Porism
  377. Invariant Area Sweep Implies Pythagorean Theorem
  378. Inversion: Reflection in a Circle
  379. Inversion in the Incircle
  380. Inversion Tool
  381. Inversion with Negative Power
  382. Invert Two Circles Into Equal Ones
  383. Isogonal and Isotomic Conjugalities
  384. Isogonal Concurrency
  385. Isogonal image of the circumcircle
  386. Isosceles on the Sides of a Triangle
  387. Isosceles Trapezoid In Cyclic Quadrilateral
  388. Isotomy and Isogonality Hand-in-Hand
  389. Iterations in Geometry, an example
  390. Iterations in Geometry, Generalization
  391. Iterations on a Circle Through Three Points
  392. Iterations on Euler Lines
  393. Johnson Circles
  394. Joseph Keech in Bride's Chair
  395. Kiepert's Centroid
  396. Kiepert's Theorem
  397. Kiepert's Triangles Graduate to Ears of Arbitrary Shape
  398. Kürschak's Tile and Theorem
  399. La Hire's Theorem
  400. La Hire's Theorem, a Variant
  401. La Hire's Theorem in Ellipse
  402. Law of Cosines
  403. Law of Cosines: PWW
  404. Law of Cosines: An Unfolded Variant
  405. Law of Cosines by Ancient Sliding
  406. Lighthouse Curves
  407. Line, Circle, and Fixed Points
  408. LCC: Apollonius' Problem with Two Circles and a Line
  409. LLC: Apollonius' Problem with Two Lines and a Circle
  410. Locus of Points in a Given Ratio to Two Points
  411. Luca Moroni's Problem In Equilateral Triangle
  412. MacLaurin's Construction of Conics
  413. Malfatti's Problem
  414. Malfatti's Problem, Hart's Solution
  415. Maximum Perimeter Property of the Incircle
  416. MathPro Logo
  417. Maxwell's Theorem
  418. Median in Touching Circles
  419. Medians in a Quadrilateral
  420. Menelaus and Ceva Theorems
  421. Menelaus From Ceva
  422. Midline in Quadrilateral
  423. Midline in Similar Triangles
  424. Midpoint Reciprocity in Napoleon's Configuration
  425. Midpoints and Orthogonality in Isosceles Triangles
  426. Midpoints from Gergonne Triangle
  427. Midpoints of the Lines Joining In- and Excenters
  428. Miguel's Tangents: Modern Day Sangaku
  429. Minkowski's addition of convex shapes
  430. Miquel's Point
  431. Miquel's Point of a 4-line Via Spiral Similarity
  432. Miquel's Theorem for Circles
  433. Mirror on the Wall
  434. Mirror property of the altitudes
  435. Mirror Property of Altitudes via Pascal's Hexagram
  436. Mixtilinear Circles and Concurrence
  437. Monge via Desargues
  438. Monge via Desargues II
  439. More Bottles in a Wine Rack
  440. Morley Constellation
  441. Morley's Equilaterals, Spiridon A. Kuruklis' proof
  442. Morley's Theorem
  443. Morley's Theorem, Cardioids and Such
  444. Morley's Theorem, G. Zsolt Kiss' proof
  445. Morley's Theorem: Taylor and Marr's Proof
  446. Nagel Line
  447. Nagel point
  448. Nagel Point of Medial Triangle
  449. Nagel's Theorem
  450. Napoleon-Barlotti Theorem
  451. Napoleon Theorem by Plane Tessellation
  452. Napoleon's Relatives
  453. Napoleon's Theorem by Transformation
  454. Neuberg Sangaku
  455. Newly Born Pair of Siblings to Archimedes' Twins
  456. Newton's Construction of Conics
  457. Newton's Theorem
  458. Nine Point Circle
  459. No Equilateral Triangles, Please
  460. Nobbs' Points, Gergonne Line
  461. Non-classic Way To Draw Cardioid
  462. No-Pedal Collinearity
  463. Not So Hidden Homotheties
  464. Novel Concurrency of Perpendicular Bisectors
  465. On Bottema's Shoulders
  466. On Bottema's Shoulders II
  467. On Bottema's Shoulders with a Ladder
  468. One More Seven Circles Theorem
  469. One Sided Surface in 4D
  470. Optical Property of Ellipse
  471. Optimal Quadrilateral Inscribed in Square, PWW
  472. Optimization Problem in Acute Angle
  473. Original Taylor and Marr's Proof
  474. Orthogonal Lines, Midpoints, and Collinearity
  475. Orthocenter and Three Equal Circles
  476. Orthocentric System From Rectangle
  477. Orthogonality in Isogonal Conjugacy
  478. Orthogonality in Isosceles Triangles
  479. Orthologic Triangles in a Quadrilateral
  480. Orthopole
  481. Pairs of Homologous Lines under Spiral Similarities
  482. Pairs of Incircles in a Quadrilateral
  483. Pantograph
  484. Pappus' Theorem
  485. Parabola As Envelope of Straight Lines
  486. Parabola As Envelope II
  487. Parabolas Related to the Orthic Triangle
  488. Parabolic Mirror, Illustration
  489. Parabolic Mirror, Theory
  490. Paragon
  491. Parahexagon
  492. Parallel Chords
  493. Parallel Chords in Ellipse
  494. Parallel Lines and Similar Triangles
  495. Parallel Lines in a Cyclic Quadrilateral
  496. Parallel Lines in Rhombus
  497. Parallel Lines in a Triangle
  498. Parallelogram and Similar Triangles
  499. Parallelogram Iterations
  500. Parallelogram Law
  501. Parallelogram Law: A PWW
  502. Parallelogram Law: A Tessellation
  503. Pascal in a Cyclic Quadrilateral
  504. Pascal Lines: Steiner and Kirkman Theorems
  505. Pascal Lines: Steiner and Kirkman Theorems II
  506. Pascal's Theorem
  507. Peaucellier Linkage
  508. Pedal Collinearities
  509. Pedal Parallelogram
  510. Pedal Polygons
  511. Pedal Triangle and Isogonal Conjugacy
  512. Pedoe's Theorem
  513. Pentagon in a Semicircle
  514. Perigal's Proof of the Pythagorean Proposition
  515. Perimeters of Parallelogram And Rhombus
  516. Perpendicular Bisectors in an Inscriptible Quadrilateral
  517. Perpendicular Bisectors in an Inscriptible Quadrilateral II
  518. Pinning Butterfly on Radical Axes
  519. Pivot Theorem
  520. Point common to two similar rectangles
  521. Polar Circle
  522. Pole and Polar with respect to a Triangle
  523. Poles and Polars
  524. Poles, Polars and Orthogonal Circles
  525. Poles, Polars and Quadrilaterals
  526. Polygon from the midpoints of its sides
  527. Polygonal Ears and Triangulation Duals
  528. Polygons Formed by Perpendicular Bisectors
  529. Poncelet's Porism
  530. Poncelet's Porism in Ellipses
  531. Poncelet's Theorem
  532. PCC: Apollonius' Problem with Two Circles and a Point
  533. PLC: Apollonius Problem with a point, a line, and a circle
  534. PPC: Apollonius' Problem with Two Points and a Circle
  535. PPL: An Elementary Solution
  536. PPL: Apollonius' Problem with Two Points and a Line
  537. Problem 4, 1975 USA Math Olympiad and the Radical Axis
  538. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
  539. Problem 4, 1975 USA Math Olympiad: Normals and Tangents
  540. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch
  541. Problem 4 from the IMO 2013 - Synthetic Solution
  542. Problem in an Equilateral Triangle
  543. Problem in an Equilateral Triangle II
  544. Problem In an Isosceles Right Triangle
  545. Problem in Three Squares
  546. Problem of Equal Steps
  547. Problem of Equal Steps II
  548. Problem of polygon construction
  549. Problem with Equilateral Triangles
  550. Product of Rotations
  551. Projections of Convex Quadrilateral
  552. Projections of Isogonal Conjugate are Concyclic
  553. Projections on Internal and External Angle Bisectors
  554. Projective Collinearity in a Quadrilateral
  555. Projective Generalization of Maxwell's Theorem
  556. Projective Proof of Pascal's Theorem
  557. Proof Without Words for the Pythagorean Theorem
  558. Properties of the Circle of Similitude
  559. Property of 6-Parpolygon
  560. Property of Angle Bisectors II
  561. Property of Circumscribed Quadrilaterals
  562. Property of Points Where In- and Excircles Touch a Triangle
  563. Property of Right Trapezoids
  564. Property of the Line IO
  565. Property of the Line IO II
  566. Property of the Line IO: a proof from the Book
  567. Property of Thébault Circles
  568. Proportions with Two Orthocenters
  569. Ptolemy on Hinges
  570. Pythagoras and Vecten Break Japan's Isolation
  571. Pythagoras' from Bottema's
  572. Pythagorean Condition in An Isosceles Right Triangle
  573. Pythagorean Theorem: Subtle Dangers of Visual Proof
  574. Pythagorean Theorem By Euclid
  575. Pythagorean Theorem By Plane Tessellation
  576. Pythagorean Theorem By Hexagonal Tessellation
  577. Pythagorean Theorem By Rearrangement
  578. Pythagorean Theorem By Shearing and Rotations I
  579. Pythagorean Theorem By Shearing and Rotations II
  580. Pythagorean Theorem in an Orthodiagonal Quadrilateral
  581. Quadrilaterals Formed by Perpendicular Bisectors
  582. Quadrilateral with 3 Equal Sides
  583. Quest for Paragon
  584. Radical Axis and Center, an Application
  585. Radical Axis of Circles Inscribed in a Circular Segment
  586. Radical Center
  587. Radius and Construction of a Mixtilinear Circle
  588. Radius of a Circle by Paper Folding
  589. Rectangle in Arbelos
  590. Rectangle Inscribed in Triangle
  591. Rectified, Halved, Sheared, Eyeballs Still Surprise
  592. Reflection in a Triangle Tangent to Parabola
  593. Reflection In Line
  594. Reflections in Ellipse
  595. Reflections of a Line Through the Orthocenter
  596. Reflections of a Point on the Circumcircle
  597. Reflections of the Orthocenter
  598. Regular Polygons in a Triangular Grid
  599. Remarkable Line in a Quadrilateral
  600. Remarkable Line in Cyclic Quadrilateral
  601. Rendezvous
  602. Surprise: Right Angle in Circle
  603. Right Isosceles Triangles on Sides of a Quadrilateral
  604. Rotation Transform
  605. Rotations in Disguise
  606. Rusty Compass Construction of Equilateral Triangle
  607. Salinon: from Archimedes Book of Lemmas
  608. Sanchez's Viviani's Area Analogue
  609. Sangaku Iterations, Is it Wasan?
  610. Sangakus with a Mixtilinear Circle
  611. Sangaku with Three Mixtilinear Circles
  612. Seven Circles Theorem
  613. Squares in Arbelos
  614. Secant, Tangents and Orthogonality
  615. See-Saw Lemma
  616. Segment Trisection
  617. Seven Equal Circles
  618. Shape of Constant Width
  619. Shearing Butterflies in Quadrilaterals
  620. Siamese Triangles
  621. Siamese Triangles II
  622. Similar Triangles on Sides of a Quadrilateral
  623. Similar Triangles on Sides and Diagonals of a Quadrilateral
  624. Simson Line
  625. Simson Line From Isogonal Perspective
  626. Simson Line in Disguise
  627. Simson Lines Of Diametrically Opposite Points
  628. Simsons and 9-Point Circles in Cyclic Quadrilateral
  629. Simultaneous Diameters in Concurrent Circles
  630. sin(N)
  631. Six Circles with Concurrent Pairwise Radical Axes
  632. Six Circles Theorem (Bui Quang Tuan)
  633. Six Circles Theorem (Elkies)
  634. Six Circles Theorem (Evelyn)
  635. Six Concyclic Circumcenters in Fermat's Configuration
  636. Six Concyclic Points
  637. Six Concyclic Points II
  638. Six Concyclic Points via Antipedal Triangle
  639. Six Incircles in an Equilateral Triangle
  640. Six Point Circle
  641. SixPoints, ThreeLines
  642. Soddy Circles and David Eppstein's Centers
  643. Some Properties of Napoleon's Configuration
  644. Spectral Decomposition of a Planar Quadrilateral: an Illustration
  645. Sperner's Lemma
  646. Spira Mirabilis - Wonderful Spiral
  647. Spiral Similarity
  648. Spiral Similarity and CollinearitySpiral Similarity and Collinearity
  649. Square from Four Points, One on Each Side
  650. Square in a Right Triangle
  651. Square Inscribed in a Triangle
  652. Square Inscribed in a Triangle II
  653. Square Inscribed In a Triangle III
  654. Square out of a Quadrilateral
  655. Square, Similarity and Slopes
  656. Squares in Circles
  657. Squares on Sides of a Parallelogram
  658. Squares on Sides of a Quadrilateral
  659. Squares on Sides of a Quadrilateral II
  660. Squinting Eyes Theorem
  661. Stan Fulger's Observation in Right Triangle
  662. Star Construction of Shapes of Constant Width
  663. Steiner's Circumellipse in Barycentric Coordinates
  664. Steiner's Chain
  665. Steiner's porism
  666. Straight Edge Construction of Polar
  667. Surprising 60° in Equilateral Triangle
  668. Symmedian and Antiparallel
  669. Symmedian and 2 Antiparallels
  670. Symmedian and the Simson
  671. Symmedian and the Tangents
  672. Symmedian in a Right Triangle
  673. Symmedian in Right Triangle II
  674. Symmedian via Parallel Transversal and Two Circles
  675. Tangency Point of Two Circles
  676. Tangent and Secant
  677. Tangent as a Radical Axis
  678. Tangent Circles and an Isosceles Triangle Tangent Circles and an Isosceles Triangle II
  679. Tangent Circles in a Parallelogram
  680. Tangent in Concurrency
  681. Tangent Lines and Circles in Convex Quadrilateral
  682. Tangents and Diagonals in Cyclic Quadrilateral
  683. Tangents, Perpendiculars and Geometric Mean
  684. Terquem's Theorem
  685. Tesseract
  686. The Midpoint of an Angle Bisector - a Problem by A. A. Zaslavsky
  687. The Plain Butterfly Theorem
  688. The Shortest Crease
  689. Thébault's Problem I
  690. Thébault's Problem II
  691. Thébault's Problem III
  692. Thébault's Problem IV
  693. Thébault's Problem III, Proof
  694. Theorem of Complete Quadrilateral
  695. Theorem of Three Tangents to a Conic
  696. Theorems of Ceva and Menelaus, an Illustrated Generalization
  697. Three Circles through the Incenter
  698. Three Concurrent Circles
  699. Three Congruent Circles by Reflection
  700. Three Congruent Circles by Reflection II
  701. Three Congruent Circles by Reflection III
  702. Three Incircles In a Right Triangle
  703. Three Incircles in a Triangle
  704. Three Isosceles Triangles
  705. Three Orthogonal Circles Through Three Given Points
  706. Three pairs of Parallel Lines and a Conic
  707. Three Parabola Tangents
  708. Three Parabolas with Common Directrix
  709. Three Parallels in a Triangle
  710. Three Points Casey's Theorem
  711. Three Points on a Parabola
  712. Three Roads, Three Travelers
  713. Three Similar Polygons
  714. Three Similar Triangles
  715. Three Similar Triangles (but differently oriented)
  716. Three Similar Triangles (but differently oriented) II
  717. Three Squares and Two Ellipses
  718. Three Tangent Circles
  719. Three Tangents Theorem
  720. Three Tangents Theorem II
  721. Three Tangents, Three Chords in Ellipse
  722. Three Tangents, Three Secants
  723. Toothpick Construction of a Square
  724. Touching Circles
  725. Touching Circles and Concurrency
  726. Touching Circles with Given Cnters II
  727. Transversal Trigonometry
  728. Triangle and Segment Count in a Polygon
  729. Triangle by HM segments
  730. Triangle Construction from Angle, Altitude and Median
  731. Triangle Geometry
  732. Triangle on a Rectangular Hyperbola
  733. Triangle Sliding on Two Straight Lines
  734. Triangles, Squares and Areas from Temple Geometry
  735. Triangles with Equal Area
  736. Triangular Billiard
  737. Trigonometric Form of Ceva's Theorem (a tool for discovery of trigonometric identities and other problems)
  738. Tromino as a Rep-tile
  739. Tucker Circles
  740. Tucker Circles Through Homothety
  741. Twin Segments in Arbelos
  742. Two Butterflies Theorem
  743. Two Butterflies Theorem II
  744. Two Butterflies Theorem III
  745. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  746. Two Circles and One More
  747. Two Circles and Two More
  748. Two Circles in a Parallelogram
  749. Two Circles Inscribed in a Parallelogram
  750. Two Circles in a Square
  751. Two Circles in a Square II
  752. Two Circles in an Angle
  753. Two Circles on Angle Bisector
  754. Two Circles on a Side of a Triangle
  755. Two Circles, Tangents and Two Collinearities
  756. Two Circumcircles And Two Pairs of Parallels
  757. Two Circumcircles in Triangle
  758. Two Congruent Circles by Reflection
  759. Two Conics, Pascal, Chasles, and Cross-Ratio
  760. Two Homotheties in a Parallelogram
  761. Two Lines - Two Circles
  762. Two Pairs of Parallel Lines in a Triangle
  763. Two Pencils of Parallel Lines
  764. Two Proofs of MenelausTheorem
  765. Two Quadrilaterals
  766. Two Quadruplets of Concyclic Points
  767. Two Quadruplets of Tangent Circles
  768. Two Rectangles in a Rectangle
  769. Two Similar Triangles with Parallel Sides
  770. Two Simsons in a Triangle
  771. Two Squares and Another Square
  772. Two Triples of Similar Triangles
  773. Two Tangents to Parabola
  774. Two Triangles Inscribed in a Conic
  775. Two Triangles Inscribed in a Conic - with Elementary Solution
  776. Two Triangles Inscribed in a Conic - with Solution
  777. Two Triangles With Common Base and Altitude
  778. Two Triples of Concurrent Cevians and a Conic
  779. Two Triples of Concurrent Circles
  780. Van Schooten's Locus Problem
  781. Varignon Parallelogram
  782. Vecten-Nikolin Hexagon
  783. Vecten's Collinearity
  784. Vecten's Mesh
  785. Vecten's Theorem
  786. Versatile Theorem
  787. Vietnamese Lemma towards an Extension of a Japanese Theorem
  788. Weitzenböck Inequality
  789. What does one see inside a spherical mirror?
  790. When a Triangle is Equilateral
  791. William Wallace's 1803 Statement of the Butterfly Theorem
  792. Wittenbauer's Parallelogram
  793. Y. Sawayama's Lemma
  794. Yet Another Concurrence on the 9-Point Circle
  795. Yet Another Seven Circles Theorem
  796. Zaslavsky's theorem

    Miscellaneous Demonstrations

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

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Copyright © 1996-2018 Alexander Bogomolny


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