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

    Arithmetic

  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

    Algebra

  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. Fixed Point of a Family of Parabolas
  89. Four Pegs That Form a Square
  90. Frieze Patterns
  91. Geometric Construction of Roots of Quadratic Equation
  92. Graph and Roots of a Third Degree Polynomial
  93. Graph and Roots of Quadratic Polynomial
  94. Graphing Equations Is Useful, II
  95. Graphing Equations Is Useful, V
  96. Heads and Tails (counting, invariance)
  97. Identities in the Multiplication Table
  98. Integer Iterations on a Circle II (superposition principle)
  99. Integer Iterations on a Circle III
  100. Integers and Rectangles
  101. Integers and Rectangles: a Proof by Induction
  102. Integers and Rectangles: Two Simple Proofs
  103. Interest Calculations
  104. Josephus Flavius Game
  105. Langford and Skolem Sequences
  106. Letter Count in a Sentence
  107. Lights Out
  108. Logistic Model
  109. Loop or Halt
  110. Mean Values
  111. Merlin's Magic Squares (modular arithmetic, boolean and linear algebra)
  112. Mini Lights Out
  113. Minimax Principle
  114. Minimax Theorem
  115. Modular Arithmetic
  116. Multiplication of Points on a Circle
  117. Multiplication of Points on an Ellipse
  118. Nine Squares
  119. Parrondo Paradox Via Simpson Paradox
  120. Peasant Multiplication
  121. Pennies in Boxes
  122. Plus or Minus (parity, invariance)
  123. PolygonalNumbers
  124. Square in a Circular Sector
  125. Proizvolov's Identity
  126. Pythagorean Triples
  127. Pythagorean Triples Calculator
  128. Quasigroup on a Triangular Grid
  129. Rectangle on a Chessboard
  130. Right Replacement
  131. Self-descriptive Strings
  132. Self-documenting Sentences
  133. Sierpinski's Gasket and Dihedral Symmetry
  134. Signs and Sums in a Sequence
  135. Signs in a Matrix
  136. Simple Cellular Automaton
  137. Simple Somos Sequence Calculator
  138. Simpson Paradox
  139. Sine of a Sum Formula
  140. Solitaire in Two Colors
  141. Squares and Circles (parity, invariance)
  142. Squares, Circles, and Triangles (modular arithmetic, invariance)
  143. Steinhaus' Problem
  144. Sum of Consecutive Integers is Triangular
  145. Sum of Consecutive Odd Numbers is Square
  146. Sum of Consecutive Triangular Numbers is Square
  147. Sums of Geometric Series - Proofs Without Words
  148. Swapping Rows and Columns
  149. Toads And Frogs Puzzle (counting, logic, problem solving)
  150. Two Colors in Two Rows
  151. Two Colors in Two Rows II
  152. Voronoi Diagram
  153. What's next?
  154. Word Problems

    Logic

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

    Fractals

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

    Probability

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

    Visual Illusions

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

    Calculus

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

    Math Magic

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

    Combinatorics

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

    Combinatorial Games

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

    Puzzles & Games

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

    Social Science

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

    Elementary Geometry

  447. 5x5 Square Grid and 5 Circes
  448. 60o Angle And Importance of Being The Other End of a Diameter
  449. Puzzle of 30 Grid Points and 5 Circles
  450. A Characterization of the Euler Line
  451. A Degenerate Case of Bottema's Configuration
  452. A Diameter As a Diagonal of Inscribed Quadrilateral
  453. A Double Meaning of an Arc's Midpoint
  454. A Family of Cyclic Quadrilaterals
  455. A Hinged Realization of a Plane Tessellation
  456. A Lemma of Equal Areas
  457. A Median Surprise
  458. A Multiplicative Identity of Areas in a Triangle
  459. A Property of a Median
  460. A Property of Equiangular Polygons
  461. A Property of Incenter Relative to Circumcircle
  462. A Property of Isogonal Lines
  463. A Property of Orthic Triangle
  464. A Property of Perspective Triangles
  465. A Straightedge Construction of the Midpoint of a Chord Common to Two Circles
  466. A Theorem of M. Mansion
  467. A Three Pegs Question
  468. A Triangle With a 45 Degrees Angle in Square
  469. Acute Angle Bisectors in a Right Triangle
  470. Al-Nayrizi's Construction
  471. An Appearance of a 30-60-90 Triangle
  472. An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem
  473. An Extra Triple of Equilateral Triangles for Napoleon
  474. An Unexpected Rhombus
  475. Angle: An Illustrated Classification
  476. Angle Bisector
  477. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise
  478. Angle Bisectors in Ellipse
  479. Angle Bisectors in Ellipse II
  480. Angle Bisectors On Circumcircle
  481. Angle Subtended by a Diameter
  482. Angles Inscribed in an Absent Circle
  483. Anticenter and Orthocenters
  484. Antiparallel
  485. Antiparallel via Three Reflections
  486. Archimedes' Law of the Lever
  487. Area of Median Triangle I
  488. Area of Median Triangle II
  489. Area of Parallelogram
  490. Area of the Intersection of Two Equal Rectangles
  491. Area of the Union of Two Squares
  492. Area of Triangle
  493. Areas between Two Parallelograms
  494. Areas in Square by Dissection
  495. Areas in Two Parallelograms and a Translation
  496. Arithmetic in Disguise
  497. Bespectacled Eyeballs Extension
  498. Between Major and Minor Circles
  499. Bottema with a Product of Rotations
  500. Bottema with Similar Triangles
  501. Bottema's Theorem - Proof Without Words
  502. Building a Bridge
  503. Building Bridges
  504. Carpet With a Hole
  505. Carpets in Triangle
  506. Center of Gravity and Motion of Material Points
  507. Chain of Four Intersecting Circles
  508. Characterization of Trapezoid
  509. Circle and Hyperbola as Lighthouse Curves II
  510. Circle Chains on Napoleon Triangles
  511. Circle Rotations and Fixed Points
  512. Circles and Semicircles in Rectangle
  513. Circle in a Square Inscribed in a Circle
  514. Circles with Equal Collinear Chords
  515. Circumcenter and Orthocenter
  516. Circumcenter on Angle Bisector
  517. Classification of Quadrilaterals
  518. Common Chord and a Tangent
  519. Concurrence in Arbelos
  520. Concurrence of Ten Nine-Point Circles
  521. Concurrence on a Circle
  522. Concurrent and Parallel Lines in Parallelogram
  523. Concyclicity from Collinearity
  524. Concyclicity in Rectangle
  525. Compass Only Construction - A Chord Tangent to an Inner circle
  526. Concyclic Points from Midpoint of an Arc
  527. Concyclicity in Two Semicircles
  528. Conic from Parallel Chords
  529. Conic in Hexagon
  530. Conic Sections as Loci of Points
  531. Conics in Triangle
  532. Construct Triangle by Angle, Altitude and Median
  533. Construct Triangle by Angle, Median, and Circumradius
  534. Construction: Parallel Line through Point
  535. Construction of a Right Triangle
  536. Construction of an Oval - the Roman Way
  537. Construction of the Angle Bisector
  538. Construction of the Perpendicular Bisector
  539. Cross Points in a Polygon
  540. Cutting a Hole in Half
  541. Cyclic Quadrilateral, Angle Bisector And Isosceles Triangle
  542. Dancing Rectangles Model Auxetic Behavior
  543. Dancing Squares or a Hinged Plane Tessellation
  544. A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
  545. Diagonal Count
  546. Dissection of a Vase
  547. Dissection of Cyclic Quadrilateral
  548. Divide a Circle into N Parts of Equal Area
  549. E. W. Dijkstra's Proof of the Pythagorean Theorem
  550. Eight Equal Tangents
  551. Equal Areas in Equilateral Triangle
  552. Equal And Perpendicular Segments in a Square
  553. Equal Areas in Regular 2n-gons
  554. Equal Chords in Crossing Circles
  555. Equal Circles, Medial Triangle and Orthocenter
  556. Equidecomposition of a Rectangle and a Square
  557. Equidecomposition of Two Rectangles
  558. Equidecomposition of a Triangle and a Rectangle
  559. Equidecomposition of a Triangle and a Rectangle II
  560. Equidecomposition of Two Parallelograms
  561. Equilateral Triangle on a Closed Curve
  562. Equilateral Triangle on Parallel Lines
  563. Equilateral Triangle on Parallel Lines II
  564. Equilateral Triangle on Three Lines
  565. Equilateral Triangles formed by Circumcenters
  566. Equilateral Triangle in Equilateral Triangle
  567. Equilic Quadrilateral I
  568. Equilic Quadrilateral I, A Variation
  569. Equilic Quadrilateral II
  570. Euclid's Segment Division
  571. Existence of the Circumcenter
  572. Existence of the Euler Line: An Elementary Proof
  573. Existence of the Orthocenter
  574. Extras in Bottema's Configuration
  575. Eye-to-Eye Theorem I
  576. Eye-to-Eye Theorem II
  577. Fibonacci Bamboozlement
  578. Famous Theorem
  579. Finding Circle Center By Ruler Alone
  580. Five Squares in Complex Numbers
  581. Fixed Point in Isosceles and Equilateral Triangles
  582. Fixed Point on Perpendicular Bisector
  583. Focal Definition of Ellipse
  584. Focus and Directrix of Ellipse
  585. Folding Square in a Line through the Center
  586. Four Circles and a Line
  587. Four Hinged Squares
  588. From Arbitrary Hexagon to Regular One
  589. From Perpendicular Center Lines to Concyclic Points
  590. From Straight Line to Regular Hexagon
  591. From Arbitrary Pentagon to Regular One
  592. Garcia's Two Circles Lemmas
  593. Geoboard
  594. Half Turn, Reflection in Point
  595. Harmonic Mean Sangaku
  596. Hinged Greek Cross Tessellation
  597. Hippocrates' Squaring of a Lune
  598. Homothety in Three Tangent Circles
  599. Homothety in Three Tangent Circles II
  600. How to Construct Tangents from a Point to a Circle
  601. Hunting Right Angles
  602. Huygens' proof of the Pythagorean Theorem
  603. In the Spirit of Thebault I
  604. Inscribed and Central Angles in a Circle
  605. Inscribed Angles
  606. Inscriptible Quadrilateral: An Illustration
  607. Integer Triangles with Two Angles in the Ratio 1 : 2
  608. Intersecting Chords Theorem - a Visual Proof
  609. Intersecting Chords Theorem - Hubert Shutrick's PWW
  610. Intersections of a Circle with the Four Quadrants
  611. Invariance in Orthodiagonal Quadrilaterals
  612. Is X a Midpoint of a Chord
  613. Isogonal Lemma
  614. Isogonal Concurrencies
  615. Isoperimetric Theorem For Quadrilaterals
  616. Isoperimetric Theorem For Quadrilaterals II
  617. Isosceles Configuration in Triangle
  618. Join Circles by Given Segment
  619. Joined Common Chords of Napoleon's Circumcircles
  620. Law of Cosines - Another PWW
  621. Lean Napoleon's Triangles
  622. Line Not Through a Center of Similarity
  623. Line Through a Center of Similarity
  624. Looking Back at Bottema
  625. Maimed Cake
  626. Mathematics in Pizzeria
  627. Mickey Might Be a Red Herring in the Mickey Mouse Theorem
  628. Midline in Trapezoid
  629. Midline in Triangle
  630. Miguel's Area of Square
  631. Miguel Ochoa's Chords And Tangents
  632. Miguel Ochoa's van Schooten is a Slanted Viviani
  633. Miguel Ochoa's van Schooten Like Theorem
  634. Miguel Ochoa's van Schooten Like Theorem II
  635. Miguel Ochoa's van Schooten Like Theorem III
  636. Miquel Circumcenters
  637. Napoleon's Hexagon
  638. Napoleon's Theorem: A third proof with complex numbers
  639. Napoleon's Theorem via Two Rotations
  640. Napoleon on Hinges
  641. Napoleon on Hinges in GeoGebra
  642. Neuberg Cubic
  643. Nine Point Circle: an Elementary Proof
  644. Octagon in Parallelogram
  645. Octagon in Parallelogram: Overlapping Regions
  646. Olof Hanner's Jigsaw Puzzle
  647. One More Property of Equilateral Triangles
  648. Orthic Semiperimeter
  649. Orthocenter and Circumcenter Are Isogonal Conjugate
  650. Orthocenters of Two Triangles Sharing Circumcenter and Base
  651. Orthodiagonal and Cyclic Quadrilaterals
  652. Orthogonality in Two Squares
  653. Pappus' Generalization of Euclid I.47
  654. Parabolic Reciprocity
  655. Parallel Chords II
  656. Parallel Chords in Crossing Circles
  657. Parallel Lines in a Quadrilateral
  658. Parallel Lines in a Quadrilateral II
  659. Parallelogram and Four Equilateral Triangles
  660. Parallelograms among Quadrilaterals
  661. Parallelogram and Ellipses
  662. Parallelogram in Four Circles
  663. Parallelogram in Parallelogram And in Projection
  664. Parallelogram with Side Lines through Fixed Points
  665. Pascal: Necessary and Sufficient
  666. Pedal Similarity
  667. Pedal Triangles of Inverse Points
  668. Perimeters of Convex Polygons, One within the Other
  669. Playing on Flanks
  670. PLL: Apollonius' Problem with Two Lines and a Point
  671. Polygon Metamorphosis
  672. Pompeiu's Theorem
  673. Praying Eyes Theorem
  674. Problem 1 from the EGMO2017
  675. Proclivity of Conjoint Archimedean Twins for Proliferation
  676. Properties of Flank Triangles - Second Proof by Symmetric Rotation
  677. Properties of Flank Triangles - Proof with an Asymetric Rotation
  678. Property of Angle Bisectors
  679. Property of Angle Bisectors III
  680. Property of Internal Angle Bisector - Hubert Shutrick's PWW
  681. Property of Rhombi
  682. Property of Right Isosceles Triangle
  683. Property of Semicircles
  684. Pure Angle Chasing
  685. Pure Angle Chasing II
  686. Pure Angle Chasing III
  687. Pythagorean Theorem By Dissection
  688. Pythagorean Theorem By Dissection II
  689. Pythagorean Theorem by Hinged Dissection
  690. Pythagorean Theorem by Hinged Dissection I
  691. Pythagorean Theorem by Hinged Dissection II
  692. Pythagorean Theorem: J. Adams' Proof by Dissection
  693. Pythagorean Theorem: Versluys' Proof by Dissection
  694. Pythagoras' Theorem By Sheer Shearing
  695. Pythagoras' Theorem By Similarity
  696. Pythagoras' Theorem Solely By Sliding
  697. Pythagorean Theorem via an Isosceles Triangle
  698. Pythagorean Segment Division
  699. Quadrilateral With Equal Opposite Sides And Angles
  700. Rectangles among Parallelograms
  701. Rectangular Areas in Circle
  702. Reflection in the Circumcenter
  703. Regular Hexagon and Triangle Joined at Vertex
  704. Reim's Similar Coins I
  705. Reim's Similar Coins II
  706. Reim's Similar Coins III
  707. Reim's Similar Coins IV
  708. Rhombus in a Cyclic Quadrilateral
  709. Right Triangles on Sides of a Square
  710. Rhombus in Circles
  711. Sangaku with Angle between a Tangent and a Chord
  712. Secant Angles
  713. Secant Angles II
  714. Secant, Normal, Tangent
  715. Segment Trisection II
  716. Segment Trisection Induced by Parallels to Medians
  717. Semi-regular Tessellation on Hinges A
  718. Semi-regular Tessellation on Hinges B
  719. Semi-regular Tessellation on Hinges C
  720. Seven Concyclic Points
  721. Shearing a Polygon into a Triangle of Equal Area
  722. Shifted by the Inradius Tangent Collinearities
  723. Similar Triangles in Crossing Circles
  724. Simple Property of Circle Through the Incenter
  725. Simple Quadrilaterals Tessellate the Plane
  726. Shearing and Translation in Pythagorean Pants
  727. Some Properties of Common Tangents
  728. Spiral Similarity Leads to Equilateral Triangle
  729. Square, 45 Degrees Angle and Pythagoras' Look-alike
  730. Square, Semicircle, and Other Curves
  731. Square From Nowhere
  732. Square in a Circle Inscribed in a Square
  733. Square in Chair
  734. Squaring a Rectangle
  735. Sum of Two Squares
  736. Symmetries in Triangle II
  737. Symmetry in an Asymmetric Configuration
  738. Tangent Circles, Similarity and Equal Segments
  739. Tangent Curiosity in Equilateral Triangle
  740. Thales' Theorem
  741. Thales on Angle Bisectors
  742. The Eutrigon Theorem
  743. The Shape of the Colosseum
  744. The Shortest Distance in a Circular Segment
  745. Three Collinear and Four Concyclic Points
  746. Three Common Chords in Three Concurrent Circles
  747. Three lines at 60 degres and Circumcenters
  748. Three Touching Circles
  749. Translated Triangles
  750. Translation Transform
  751. Triangle Areas in a Parallelogram
  752. Triangle Areas in a Parallelogram II
  753. Triangle Classification
  754. Trigonometric Functions
  755. Triangular Areas in Circle
  756. Twin Segments in Four Semicircles
  757. Two Altitudes, One Midpoint
  758. Two Color Coloring of the Plane
  759. Two Common Chords in Three Concurrent Circles
  760. Two Diameters and Longest Common Chord
  761. Two Equilateral Triangles
  762. Two Intersecting Circles
  763. Two Parallels in a Triangle and One More
  764. Two Pascals Merge into One
  765. Two Properties of Flank Triangles - First Proof by Symmetric Rotation
  766. Two Properties of Flank Triangles - A Proof with Complex Numbers
  767. Two Properties of Flank Triangles - and a Third One
  768. Two Regular Pentagons Joined at Vertex
  769. Two Symmetric Triangles Are Directly Equidecomposable
  770. Two Symmetric Triangles Are Directly Equidecomposable II
  771. Two Symmetric Triangles Are Directly Equidecomposable III
  772. Two Symmetric Triangles Are Directly Equidecomposable IV
  773. Two Touching Circles
  774. Van Khea's Quickie
  775. Van Schooten's and Pompeiu's Theorems
  776. Viviani's Theorem
  777. Viviani's Theorem II
  778. Viviani in Isosceles Triangle
  779. Yet Another Generalization of Bottema's Theorem

    Stereometry

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

    Outline (Fill-It-In) Mathematics

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

    Geometric Fallacies

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

 

    Geometry

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

    Miscellaneous Demonstrations

  802. Calculation of the Digits of pi by the Spigot Algorithm
  803. Calculation of the Digits of pi (Faster Version)
  804. Convex Polygon Is the Intersection of Half Planes
  805. Disapperances à la Sam Loyd
  806. Disappearing Lines
  807. Disappearing Rectangles
  808. Distance Between Strings
  809. Experimentation with Dynamic Geometry Software: An Example
  810. Genetic Algorithm Solves the Toads and Frogs Puzzle
  811. Impossible Fork
  812. Impossible Frame
  813. Morphing
  814. Order and Chaos in Multiple Pendulums
  815. Projectile Trajectory
  816. Random Dot Stereograms
  817. Ratchet Effect
  818. Rotating Gears
  819. Sorting Algorithms
  820. Structural Constellation
  821. Watt's and Chebyshev's Linkage
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Copyright © 1996-2018 Alexander Bogomolny

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