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

    Logic

  145. Lewis Carroll's Bilateral Diagrams
  146. Lewis Carroll's Trilateral Diagrams
  147. Lewis Carroll's Logic Game
  148. Venn Diagrams I
  149. Venn Diagrams II
  150. Formal Systems - an Example

    Fractals

  151. All Peano Curves
  152. Apollonian Gasket
  153. Color Cycling on the Mandelbrot Set
  154. Cantor sets
  155. Dot Patterns, Pascal Triangle and Lucas Theorem
  156. Emergence of Chaos
  157. Following the Hilbert Curve
  158. Fractal Curves and Dimension
  159. Iterated Function Systems
  160. Iterations in the Mandelbrot Set
  161. Koch's Snowflake
  162. Mandelbrot Set and Indexing of Julia Sets
  163. Plane Filling Curves: Hilbert's and Moore's
  164. Plane Filling Curves: One of Sierpinski's Curves
  165. Plane Filling Curves: the Lebesgue Curve
  166. Plane Filling Curves: Peano's and Wunderlich's
  167. Sierpinski Gasket By Common Trema Removal
  168. Sierpinski Gasket Via Chaos Game
  169. Sierpinski Gasket by Trema Removal
  170. The Chaos Game: Address Space vs IFS
  171. Variations on the Theme of Tremas
  172. Variations on the Theme of Tremas II

    Probability

  173. Bertrand's Paradox
  174. Binomial Distribution
  175. Birds On a Wire
  176. Buffon's Noodle
  177. Chess Players Truel
  178. Conditional Probability and Independent Events
  179. Eight Selections in Six Sectors
  180. Family Statistics
  181. Lewis Carroll's pillow problem
  182. Monty Hall Dilemma's Simulation
  183. Parrondo Paradox
  184. Random Clock Hands
  185. Stick Broken Into Three Pieces (Cartesian Coordinates)
  186. Stick Broken Into Three Pieces (Trilinear Coordinates)
  187. Who Needs Monty?

    Visual Illusions

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

    Calculus

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

    Math Magic

  261. An Arithmetic Magic Trick
  262. Arithmetic magic matrix
  263. Bachet's Magic Trick
  264. Barcode Magic
  265. Calendar Magic
  266. Gergonne's Magic Trick
  267. Hummer's Mind Reader
  268. Magic in Square
  269. Math Telepathy
  270. Number Guessing Game
  271. Two Numbers Guessing Game

    Combinatorics

  272. 3-Colors Tower of Hanoi
  273. Affirmative Action Problem
  274. Balanced Incomplete Block Design on a Triangular Grid
  275. Ballot Lemma
  276. Bicolor Tower of Hanoi
  277. Blithe 12
  278. Cheapest Link & Kruskal's Algorithms - skill practice
  279. Chvatal's Art Gallery Theorem
  280. Coin change
  281. Coloring a Graph
  282. Construction of Pascal's Triangle
  283. Critical Path in a Weighted Digraph
  284. Scheduling and Critical Path Algorithm
  285. de Bruijn Cycle
  286. Dominoes on a Chessboard
  287. Euler & Hamilton Paths - skill practice
  288. Filling a Grid with Good Neighbors
  289. Five Lattice Points
  290. Fleury's Algorithm and Euler's Paths and Cycles
  291. Flipping Items Simultaneously II
  292. Flipping Items Simultaneously III
  293. Flipping pancakes
  294. Futurama Theorem and Puzzle
  295. Gray Codes
  296. Group multiplication of permutations
  297. Finding Hamilton Paths and Circuits
  298. Hanoing
  299. Happy 8
  300. Hex Can't End in a Draw
  301. How to define a permutation
  302. Infinite Latin Square
  303. Jumping Isn't Everything
  304. Latin Squares
  305. Leibniz and Pascal Triangles
  306. Life-like Automaton With Definable Rules
  307. Light Bulbs in a Circle
  308. Listing All Combinations
  309. Listing All Permutations
  310. Listing All Permutations: Johnson-Trotter Method
  311. Lucky 7
  312. Make Your Move, Kid!
  313. Marriage Problem, simulation
  314. Mazes
  315. Minumum Spanning Trees and Kruskal's Algorithm
  316. Monotone Number Sequences
  317. No Flipping without End
  318. Number sequences and Pigeonhole
  319. Party Acquaintances
  320. Permutations as a Product of Transpositions
  321. Product of Permutations
  322. Project Scheduling
  323. Property of the Powers of 2
  324. Random Walks
  325. Recombining Permutations
  326. Reflection Lemma
  327. Sam Loyd's Fifteen
  328. Scrub Tile Puzzle
  329. Shuttle Puzzle
  330. Simple Graphs
  331. Simple Graph Practice
  332. Sliders (slider games on torus, Moebius strip, projective plane, permutations, cycles)
  333. Straight Tromino on a Chessboard
  334. The Game of Life
  335. Tiling Rectangles with L-Trominoes
  336. Tool for Constructing Orthogonal Latin Squares
  337. Tower of Hanoi
  338. Tower of Hanoi, the Hard Way
  339. Traveling Salesman Problem
  340. Two Men of Tibet
  341. Universal Coloring
  342. Y Can't End in a Draw

    Combinatorial Games

  343. Date Game
  344. Dawson's Chess
  345. Dawson's Kayles
  346. Fraction Game
  347. Grundy's Game
  348. Hex 7
  349. Kayles
  350. Nim
  351. Nimble
  352. Northcott's game
  353. Odd Scoring
  354. One Pile
  355. Plainim
  356. Plainim Misère
  357. Scoring
  358. Scoring Misère
  359. Scoring Misère: Two Heaps Perfect Strategy
  360. Silver Dollar Game
  361. Silver Dollar Game With No Silver Dollar
  362. Sticks, Jess realization of Scoring Misère
  363. Sticky Problem
  364. Sticky Problem II
  365. Subtraction Game
  366. TacTix
  367. Take-Away Games
  368. Turning Turtles
  369. Wythoff's Nim
  370. Wythoff's Nim II

    Puzzles & Games

  371. 3-Chain
  372. 4×4×4 Tic Tac Toe (spacial visualization)
  373. A Game of Candy Squares
  374. Achieve the Equilibrium
  375. Bulls and Cows
  376. Caterpillar
  377. Changing Colors (combinatorics, invariance)
  378. Changing Colors II (invariance)
  379. Color Balls
  380. Computer Mastermind (logic and problem solving)
  381. Covering a Chessboard with a Hole with L-Trominoes
  382. Eight Digits Puzzle
  383. Escape of the Clones
  384. Farmer and Wife To Catch Rooster and Hen (invariance, city block distance)
  385. Fif (multiple representations)
  386. Fill a Hexagonal Grid (space visualization)
  387. Fish Soup Game (multiple representations)
  388. Four Knights (multiple representations)
  389. Full House
  390. Getting a Scout out of Desert
  391. Goat, Cabbage and Wolf to move across a river (multiple representations)
  392. Hare and Hounds
  393. Jam (multiple representations)
  394. Looping Chips (multiple representations)
  395. Mastermind (logic and problem solving)
  396. Mastermind Variants (logic and problem solving)
  397. Mirror Maze
  398. Peg Solitaire (logic, problem solving, subproblems)
  399. Pick A Label (logic, problem solving)
  400. Reverse Solitaire (logic, problem solving, subproblems)
  401. Sending scouts into the desert (algebra, geometric series, problem solving)
  402. Seven Coins Puzzle (multiple representations)
  403. Shuttle Puzzle Practice I
  404. Shuttle Puzzle Practice II
  405. Skyscrapers Puzzle
  406. Solitaire on a Circle (iterations, invariance)
  407. Stones in Cups
  408. Super Nim
  409. Thugs and Bowmen
  410. Tiling a Triangulated Hexagon (a proof without words)
  411. Toads And Frogs in 2 dimensions (repeated application of same idea)
  412. Tromino Puzzle: Golomb's Theorem
  413. Tromino Puzzle: Deficient Squares
  414. Vector Addition and Subtraction
  415. Vowel Movement
  416. Wolf and Rabbit

    Geometric Fallacies

  417. A Circle With Two Centers
  418. All Triangles Are Isosceles
  419. Curry's Paradox
  420. Delian Problem Solved
  421. A Faulty Dissection
  422. Hooper's Paradox
  423. Is Every Trapezoid Parallelogram?
  424. John Sharp's Paradox
  425. Langman's Paradox
  426. Popping A Square
  427. Rouse Ball's Fallacy
  428. Sam Loyd's Son's Dissection
  429. Two Perpendiculars From a Point to a Line

    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. A Characterization of the Euler Line
  449. A Degenerate Case of Bottema's Configuration
  450. A Diameter As a Diagonal of Inscribed Quadrilateral
  451. A Double Meaning of an Arc's Midpoint
  452. A Hinged Realization of a Plane Tessellation
  453. A Lemma of Equal Areas
  454. A Multiplicative Identity of Areas in a Triangle
  455. A Property of Equiangular Polygons
  456. A Three Pegs Question
  457. Acute Angle Bisectors in a Right Triangle
  458. Al-Nayrizi's Construction
  459. An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem
  460. Angle: An Illustrated Classification
  461. Angle Bisector
  462. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise
  463. Angle Bisectors in Ellipse
  464. Angle Bisectors in Ellipse II
  465. Angle Bisectors On Circumcircle
  466. Angle Subtended by a Diameter
  467. Angles Inscribed in an Absent Circle
  468. Antiparallel
  469. Antiparallel via Three Reflections
  470. Archimedes' Law of the Lever
  471. Area of Median Triangle
  472. Area of Parallelogram
  473. Area of the Union of Two Squares
  474. Area of Triangle
  475. Areas in Square by Dissection
  476. Arithmetic in Disguise
  477. Between Major and Minor Circles
  478. Bottema's Theorem - Proof Without Words
  479. Building a Bridge
  480. Building Bridges
  481. Carpet With a Hole
  482. Circle and Hyperbola as Lighthouse Curves II
  483. Circle Chains on Napoleon Triangles
  484. Circles and Semicircles in Rectangle
  485. Circle in a Square Inscribed in a Circle
  486. Circles with Equal Collinear Chords
  487. Circumcenter on Angle Bisector
  488. Classification of Quadrilaterals
  489. Common Chord and a Tangent
  490. Concyclicity in Rectangle
  491. Compass Only Construction - A Chord Tangent to an Inner circle
  492. Conic Sections as Loci of Points
  493. Construction of a Right Triangle
  494. Construction of an Oval - the Roman Way
  495. Construction of the Angle Bisector
  496. Construction of the Perpendicular Bisector
  497. Cross Points in a Polygon
  498. Dancing Rectangles Model Auxetic Behavior
  499. Dancing Squares or a Hinged Plane Tessellation
  500. A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
  501. Diagonal Count
  502. Dissection of a Vase
  503. Dissection of Cyclic Quadrilateral
  504. Divide a Circle into N Parts of Equal Area
  505. E. W. Dijkstra's Proof of the Pythagorean Theorem
  506. Eight Equal Tangents
  507. Equal Areas in Equilateral Triangle
  508. Equal Areas in Regular 2n-gons
  509. Equal Chords in Crossing Circles
  510. Equal Circles, Medial Triangle and Orthocenter
  511. Equidecomposition of a Rectangle and a Square
  512. Equidecomposition of Two Rectangles
  513. Equidecomposition of a Triangle and a Rectangle
  514. Equidecomposition of a Triangle and a Rectangle II
  515. Equidecomposition of Two Parallelograms
  516. Equilateral Triangle on Parallel Lines
  517. Equilic Quadrilateral I
  518. Equilic Quadrilateral I, A Variation
  519. Equilic Quadrilateral II
  520. Euclid's Segment Division
  521. Existence of the Circumcenter
  522. Existence of the Euler Line: An Elementary Proof
  523. Fibonacci Bamboozlement
  524. Famous Theorem
  525. Finding Circle Center By Ruler Alone
  526. Fixed Point in Isosceles and Equilateral Triangles
  527. Focal Definition of Ellipse
  528. Focus and Directrix of Ellipse
  529. Folding Square in a Line through the Center
  530. Four Hinged Squares
  531. Geoboard
  532. Half Turn, Reflection in Point
  533. Harmonic Mean Sangaku
  534. Hinged Greek Cross Tessellation
  535. Hippocrates' Squaring of a Lune
  536. How to Construct Tangents from a Point to a Circle
  537. Hunting Right Angles
  538. Huygens' proof of the Pythagorean Theorem
  539. In the Spirit of Thebault I
  540. Inscribed and Central Angles in a Circle
  541. Inscribed Angles
  542. Inscriptible Quadrilateral: An Illustration
  543. Intersecting Chords Theorem - a Visual Proof
  544. Intersecting Chords Theorem - Hubert Shutrick's PWW
  545. Intersections of a Circle with the Four Quadrants
  546. Invariance in Orthodiagonal Quadrilaterals
  547. Join Circles by Given Segment
  548. Law of Cosines - Another PWW
  549. Lean Napoleon's Triangles
  550. Maimed Cake
  551. Mathematics in Pizzeria
  552. Midline in Trapezoid
  553. Midline in Triangle
  554. Napoleon's Theorem: A third proof with complex numbers
  555. Napoleon's Theorem via Two Rotations
  556. Napoleon on Hinges
  557. Neuberg Cubic
  558. Nine Point Circle: an Elementary Proof
  559. Octagon in Parallelogram
  560. Octagon in Parallelogram: Overlapping Regions
  561. Olof Hanner's Jigsaw Puzzle
  562. Orthic Semiperimeter
  563. Orthocenters of Two Triangles Sharing Circumcenter and Base
  564. Pappus' Generalization of Euclid I.47
  565. Orthodiagonal and Cyclic Quadrilaterals
  566. Parallel Chords II
  567. Parallel Chords in Crossing Circles
  568. Parallel Lines in a Quadrilateral
  569. Parallelogram and Ellipses
  570. Pascal: Necessary and Sufficient
  571. PLL: Apollonius' Problem with Two Lines and a Point
  572. Polygon Metamorphosis
  573. Pompeiu's Theorem
  574. >Two Properties of Flank Triangles - First Proof by Symmetric Rotation
  575. Properties of Flank Triangles - Second Proof by Symmetric Rotation
  576. Properties of Flank Triangles - Proof with an Asymetric Rotation
  577. Two Properties of Flank Triangles - A Proof with Complex Numbers
  578. Two Properties of Flank Triangles - and a Third One
  579. Property of Angle Bisectors
  580. Property of Angle Bisectors III
  581. Property of Internal Angle Bisector - Hubert Shutrick's PWW
  582. Property of Rhombi
  583. Property of Semicircles
  584. Pythagorean Theorem By Dissection
  585. Pythagorean Theorem By Dissection II
  586. Pythagorean Theorem by Hinged Dissection
  587. Pythagorean Theorem by Hinged Dissection I
  588. Pythagorean Theorem by Hinged Dissection II
  589. Pythagorean Theorem: J. Adams' Proof by Dissection
  590. Pythagorean Theorem: Versluys' Proof by Dissection
  591. Pythagoras' Theorem By Sheer Shearing
  592. Pythagoras' Theorem By Similarity
  593. Pythagoras' Theorem Solely By Sliding
  594. Pythagorean Theorem via an Isosceles Triangle
  595. Pythagorean Segment Division
  596. Quadrilateral With Equal Opposite Sides And Angles
  597. Rhombus in a Cyclic Quadrilateral
  598. Right Triangles on Sides of a Square
  599. Sangaku with Angle between a Tangent and a Chord
  600. Secant Angles
  601. Secant Angles II
  602. Secant, Normal, Tangent
  603. Segment Trisection Induced by Parallels to Medians
  604. Semi-regular Tessellation on Hinges A
  605. Semi-regular Tessellation on Hinges B
  606. Semi-regular Tessellation on Hinges C
  607. Shearing a Polygon into a Triangle of Equal Area
  608. Simple Quadrilaterals Tessellate the Plane
  609. Shearing and Translation in Pythagorean Pants
  610. Square From Nowhere
  611. Square in a Circle Inscribed in a Square
  612. Square in Chair
  613. Squaring a Rectangle
  614. Thales' Theorem
  615. The Eutrigon Theorem
  616. The Shape of the Colosseum
  617. Three Touching Circles
  618. Translated Triangles
  619. Translation Transform
  620. Triangle Areas in a Parallelogram
  621. Triangle Areas in a Parallelogram II
  622. Triangle Classification
  623. Trigonometric Functions
  624. Two Altitudes, One Midpoint
  625. Two Color Coloring of the Plane
  626. Two Equilateral Triangles
  627. Two Intersecting Circles
  628. Two Parallels in a Triangle and One More
  629. Two Symmetric Triangles Are Directly Equidecomposable
  630. Two Symmetric Triangles Are Directly Equidecomposable II
  631. Two Symmetric Triangles Are Directly Equidecomposable III
  632. Two Symmetric Triangles Are Directly Equidecomposable IV
  633. Two Touching Circles
  634. Van Schooten's and Pompeiu's Theorems
  635. Viviani's Theorem
  636. Viviani's Theorem II
  637. Viviani in Isosceles Triangle

    Miscellaneous Demonstrations

  638. Calculation of the Digits of pi by the Spigot Algorithm
  639. Calculation of the Digits of pi (Faster Version)
  640. Convex Polygon Is the Intersection of Half Planes
  641. Disapperances à la Sam Loyd
  642. Disappearing Lines
  643. Disappearing Rectangles
  644. Distance Between Strings
  645. Experimentation with Dynamic Geometry Software: An Example
  646. Genetic Algorithm Solves the Toads and Frogs Puzzle
  647. Impossible Fork
  648. Impossible Frame
  649. Morphing
  650. Order and Chaos in Multiple Pendulums
  651. Projectile Trajectory
  652. Random Dot Stereograms
  653. Ratchet Effect
  654. Rotating Gears
  655. Sorting Algorithms
  656. Structural Constellation
  657. Watt's and Chebyshev's Linkage

    Outline (Fill-It-In) Mathematics

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

 

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

    Stereometry

  656. Angles in a Cube I
  657. Angles in a Cube II
  658. Angles in a Cube III
  659. Bimedians in a Regular Tetrahedron
  660. Cut the Cone
  661. Cut the Cube
  662. Cut the Cylinder
  663. Knots on a Torus
  664. Möbius' Strip
  665. Multifaceted Cork
  666. One Sheet Hyperboloid
  667. Platonic Solids - Regular Polyhedra
  668. Other Solids

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