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. Multiplication of Points on a Circle
  111. Multiplication of Points on an Ellipse
  112. Nine Squares
  113. Parrondo Paradox Via Simpson Paradox
  114. Peasant Multiplication
  115. Pennies in Boxes
  116. Plus or Minus (parity, invariance)
  117. PolygonalNumbers
  118. Proizvolov's Identity
  119. Pythagorean Triples
  120. Pythagorean Triples Calculator
  121. Quasigroup on a Triangular Grid
  122. Rectangle on a Chessboard
  123. Right Replacement
  124. Self-descriptive Strings
  125. Self-documenting Sentences
  126. Sierpinski's Gasket and Dihedral Symmetry
  127. Signs and Sums in a Sequence
  128. Signs in a Matrix
  129. Simple Cellular Automaton
  130. Simple Somos Sequence Calculator
  131. Simpson Paradox
  132. Sine of a Sum Formula
  133. Solitaire in Two Colors
  134. Squares and Circles (parity, invariance)
  135. Squares, Circles, and Triangles (modular arithmetic, invariance)
  136. Steinhaus' Problem
  137. Sum of Consecutive Integers is Triangular
  138. Sum of Consecutive Odd Numbers is Square
  139. Sum of Consecutive Triangular Numbers is Square
  140. Sums of Geometric Series - Proofs Without Words
  141. Swapping Rows and Columns
  142. Toads And Frogs Puzzle (counting, logic, problem solving)
  143. Two Colors in Two Rows
  144. Two Colors in Two Rows II
  145. Voronoi Diagram
  146. What's next?
  147. Word Problems
  148. Yet Another Generalization of Bottema's Theorem

    Logic

  149. Lewis Carroll's Bilateral Diagrams
  150. Lewis Carroll's Trilateral Diagrams
  151. Lewis Carroll's Logic Game
  152. Venn Diagrams I
  153. Venn Diagrams II
  154. Formal Systems - an Example

    Fractals

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

    Probability

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

    Visual Illusions

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

    Calculus

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

    Math Magic

  266. An Arithmetic Magic Trick
  267. Arithmetic magic matrix
  268. Bachet's Magic Trick
  269. Barcode Magic
  270. Calendar Magic
  271. Gergonne's Magic Trick
  272. Hummer's Mind Reader
  273. Magic in Square
  274. Math Telepathy
  275. Number Guessing Game
  276. Two Numbers Guessing Game

    Combinatorics

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

    Combinatorial Games

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

    Puzzles & Games

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

    Geometric Fallacies

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

    Social Science

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

    Elementary Geometry

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

    Outline (Fill-It-In) Mathematics

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

    Stereometry

  753. Angles in a Cube I
  754. Angles in a Cube II
  755. Angles in a Cube III
  756. Bimedians in a Regular Tetrahedron
  757. Cut the Cone
  758. Cut the Cube
  759. Cut the Cylinder
  760. Knots on a Torus
  761. Möbius' Strip
  762. Multifaceted Cork
  763. One Sheet Hyperboloid
  764. Platonic Solids - Regular Polyhedra
  765. Other Solids

 

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

    Miscellaneous Demonstrations

  758. Calculation of the Digits of pi by the Spigot Algorithm
  759. Calculation of the Digits of pi (Faster Version)
  760. Convex Polygon Is the Intersection of Half Planes
  761. Disapperances à la Sam Loyd
  762. Disappearing Lines
  763. Disappearing Rectangles
  764. Distance Between Strings
  765. Experimentation with Dynamic Geometry Software: An Example
  766. Genetic Algorithm Solves the Toads and Frogs Puzzle
  767. Impossible Fork
  768. Impossible Frame
  769. Morphing
  770. Order and Chaos in Multiple Pendulums
  771. Projectile Trajectory
  772. Random Dot Stereograms
  773. Ratchet Effect
  774. Rotating Gears
  775. Sorting Algorithms
  776. Structural Constellation
  777. Watt's and Chebyshev's Linkage

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