Memory and Matching
 Match answers
 Memory  Easy
 Memory  Medium
 Memory  Hard
Arithmetic
 1089 and a Property of 3digit Numbers
 2 Pails Puzzle
 3 Jugs Puzzle
 3 Jugs Puzzle in Barycentric Coordinates
 3Term Arithmetic Progression
 Abacus in Various Number Systems
 Algorithm for Computing LCM
 Aliquot Game
 Arithmetic Operations in Various Bases
 Broken Calculator
 Coin Game
 Collatz Conjecture
 Common Multiples and the Least Common Multiple
 Compare Fractions: Interactive Practice
 Counting and Grouping
 Distributive Law
 EAN13 Encoding
 Equivalent Fractions
 Euclid's Algorithm: An Interactive Illustration
 Euclid's Game
 Euclid's Game on a Square Grid
 Even, Odd and Total Number of Digits
 Factoring with the Factor Tree
 Farey Fractions
 Fibonacci Numbers in Equilateral Triangle
 Flipping Items Simultaneously
 Fraction Comparison
 Fraction Comparison Sped up
 GCD and LCM via Factor Tree
 GCD and LCM by Plain Factorization
 GCD(M, N) × LCM(M, N) = M × N
 Geometric view of the Sieve of Eratosthenes
 ISBN Encoding
 Kaprekar's Iterations and Numbers
 Lattice Multiplication
 Long Multiplication
 Less, Equal, More
 Listing All the Composite Numbers
 Lucky Numbers
 Make An Identity
 Make It All Zeros
 Moving Chips in Pairs Down a Checkerboard
 Napier Bones
 Plain Multiplication table up to 20×20
 Product of Simple Fractions
 Rule of Change of Addends
 Sieve of Eratosthenes
 Sieve of Squares
 Soroban in Various Number Systems
 Suan Pan in Various Number Systems
 Subtraction in a Rectangular Array
 Sums and Products
 Sums of Powers of Digits
 Treasure Hunt in a Square Grid
 Treasure Hunt from Outside the Grid
 Two Simple Equations
 Universal Product Code
 Zeros and Nines
Algebra
 A Line in a Square Grid
 ab+bc+ca ≤ aa+bb+cc
 Ambassadors at a Round Table
 An Arctan Identity and its Consequence
 Averages, Arithmetic and Harmonic Means
 An Inequality with Complex Numbers of Unit Length
 An Inequality with Complex Numbers of Unit Length II
 Base (Binary, Decimal, etc.) Converter
 Binary Color Device
 Breaking Chocolate Bars
 Candy Game: Integer Iterations on a Circle
 Candy Game (Change Discharged)
 Chebyshev polynomials
 Complex Numbers: A Dynamic Tool
 Constructible Numbers
 Counting Chips On a Circle
 Counting Squares in a Square
 Counting Triangles
 Counting Triangles II
 Divisibility Problem
 Dots and Fractions
 Ducci Sequences
 Egyptian Multiplication
 Filling an Array with 0s and 1s  and Counting
 First Geometric Interpretation of Complex Numbers
 Four Pegs That Form a Square
 Frieze Patterns
 Geometric Construction of Roots of Quadratic Equation
 Graph and Roots of a Third Degree Polynomial
 Graph and Roots of Quadratic Polynomial
 Graphing Equations Is Useful, II
 Graphing Equations Is Useful, V
 Heads and Tails (counting, invariance)
 Identities in the Multiplication Table
 Integer Iterations on a Circle II (superposition principle)
 Integer Iterations on a Circle III
 Integers and Rectangles
 Integers and Rectangles: a Proof by Induction
 Integers and Rectangles: Two Simple Proofs
 Interest Calculations
 Josephus Flavius Game
 Langford and Skolem Sequences
 Letter Count in a Sentence
 Lights Out
 Logistic Model
 Loop or Halt
 Mean Values
 Merlin's Magic Squares (modular arithmetic, boolean and linear algebra)
 Mini Lights Out
 Minimax Principle
 Minimax Theorem
 Modular Arithmetic
 Multiplication of Points on a Circle
 Multiplication of Points on an Ellipse
 Nine Squares
 Parrondo Paradox Via Simpson Paradox
 Peasant Multiplication
 Pennies in Boxes
 Plus or Minus (parity, invariance)
 PolygonalNumbers
 Proizvolov's Identity
 Pythagorean Triples
 Pythagorean Triples Calculator
 Quasigroup on a Triangular Grid
 Rectangle on a Chessboard
 Right Replacement
 Selfdescriptive Strings
 Selfdocumenting Sentences
 Sierpinski's Gasket and Dihedral Symmetry
 Signs and Sums in a Sequence
 Signs in a Matrix
 Simple Cellular Automaton
 Simple Somos Sequence Calculator
 Simpson Paradox
 Sine of a Sum Formula
 Solitaire in Two Colors
 Squares and Circles (parity, invariance)
 Squares, Circles, and Triangles (modular arithmetic, invariance)
 Steinhaus' Problem
 Sum of Consecutive Integers is Triangular
 Sum of Consecutive Odd Numbers is Square
 Sum of Consecutive Triangular Numbers is Square
 Sums of Geometric Series  Proofs Without Words
 Swapping Rows and Columns
 Toads And Frogs Puzzle (counting, logic, problem solving)
 Two Colors in Two Rows
 Two Colors in Two Rows II
 Voronoi Diagram
 What's next?
 Word Problems
Logic
 Lewis Carroll's Bilateral Diagrams
 Lewis Carroll's Trilateral Diagrams
 Lewis Carroll's Logic Game
 Venn Diagrams I
 Venn Diagrams II
 Formal Systems  an Example
Fractals
 All Peano Curves
 Apollonian Gasket
 Color Cycling on the Mandelbrot Set
 Cantor sets
 Dot Patterns, Pascal Triangle and Lucas Theorem
 Emergence of Chaos
 Exercise with Square Spiral
 Following the Hilbert Curve
 Fractal Curves and Dimension
 Iterated Function Systems
 Iterations in the Mandelbrot Set
 Koch's Snowflake
 Mandelbrot Set and Indexing of Julia Sets
 Plane Filling Curves: Hilbert's and Moore's
 Plane Filling Curves: One of Sierpinski's Curves
 Plane Filling Curves: the Lebesgue Curve
 Plane Filling Curves: Peano's and Wunderlich's
 Sierpinski Gasket By Common Trema Removal
 Sierpinski Gasket Via Chaos Game
 Sierpinski Gasket by Trema Removal
 The Chaos Game: Address Space vs IFS
 Variations on the Theme of Tremas
 Variations on the Theme of Tremas II
Probability
 Bertrand's Paradox
 Binomial Distribution
 Birds On a Wire
 Buffon's Noodle
 Chess Players Truel
 Conditional Probability and Independent Events
 Eight Selections in Six Sectors
 Family Statistics
 Lewis Carroll's pillow problem
 Monty Hall Dilemma's Simulation
 Parrondo Paradox
 Random Clock Hands
 Stick Broken Into Three Pieces (Cartesian Coordinates)
 Stick Broken Into Three Pieces (Trilinear Coordinates)
 Who Needs Monty?
Visual Illusions
 Assimilation Illusion
 Balls and Shadows Illusion
 Brick Wall Illusion
 Bulging Lines illusion
 Circular Poggendorff Illusion
 Continuous Line Illusion
 Cube Counting Ambiguity
 Day's Sine Illusion
 Delboeuf Illusion
 Distorted Lines Illusion
 Divergent Spirals Illusion
 Gerbino's Illusion
 Heaver: A Visual Illusion
 Height/Width Interplay Illusion
 Hering Illusion
 Hermann Grid Illusion
 Importance of Occluders: Motion Binding
 Judd Illusion
 Kanizsa Triangle
 MüllerLyer Illusion
 Necker Cube
 Occluded Wheel Illusion
 Orbison Illusion
 Parallelogram Illusion
 Perspective Illusion
 Poggendorff Illusion
 Poggendorff Illusion II
 Ponzo Illusion
 Ponzo Illusion II
 Reverse Spoke Illusion
 Revolving Circles Illusion
 Rotating Top Illusion
 Scintillating Grid Illusion
 Shepard's Parallelogram Illusion
 Shifting Cube Illusion
 Shimmer Illusion
 Simultaneous Contrast Illusion
 Simultaneous Contrast Illusion II
 Simultaneous Contrast: Koffka's Illusion
 Simultaneous Contrast: White's Illusion
 Simultaneous Contrast: White's Illusion, VVariant
 Staircase Illusion
 Twisted Cord Illusion
 Unfolding Spirals Illusion
 Wundt Illusion
 Wundt Block Illusion
 Zöllner's Illusion
Calculus
 Addition and Subtraction of Functions
 Area of a Circle by Leonardo da Vinci
 Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
 Cartesian Coordinate System
 Cavalieri's Principle
 Derivative of Sine and Cosine
 Distance From a Point to a Straight Line
 Equations of a Straight Line
 Estimating Circumference of a Circle
 Focal Definition of Parabola
 Function, Derivative and Integral
 Graph of a Polynomial
 Graph of a Polynomial Defined by Its Roots
 Horner's Method
 Inflection Points of Fourth Degree Polynomials
 Lagrange Interpolation
 Linear Function with Coefficients in Arithmetic Progression
 Maximum Volume of a Cut Off Box
 Mistrust Intuition of the Infinite
 Naturally Discontinuous Functions
 Newton's Method
 Riemann Sums  Function Integration
 Rolle's and Mean Value Theorems
 Roots and Tangents
 Schwarz Lantern
 Taylor Series Approximation to Cosine
 Two Circles, Two Segments  One Ratio
 Two Circles and a Limit
Math Magic
 An Arithmetic Magic Trick
 Arithmetic magic matrix
 Bachet's Magic Trick
 Barcode Magic
 Calendar Magic
 Gergonne's Magic Trick
 Hummer's Mind Reader
 Magic in Square
 Math Telepathy
 Number Guessing Game
 Two Numbers Guessing Game
Combinatorics
 3Colors Tower of Hanoi
 Affirmative Action Problem
 Balanced Incomplete Block Design on a Triangular Grid
 Ballot Lemma
 Bicolor Tower of Hanoi
 Blithe 12
 Cheapest Link & Kruskal's Algorithms  skill practice
 Chvatal's Art Gallery Theorem
 Coin change
 Coloring a Graph
 Construction of Pascal's Triangle
 Critical Path in a Weighted Digraph
 Scheduling and Critical Path Algorithm
 de Bruijn Cycle
 Dominoes on a Chessboard
 Euler & Hamilton Paths  skill practice
 Filling a Grid with Good Neighbors
 Five Lattice Points
 Fleury's Algorithm and Euler's Paths and Cycles
 Flipping Items Simultaneously II
 Flipping Items Simultaneously III
 Flipping pancakes
 Futurama Theorem and Puzzle
 Gray Codes
 Group multiplication of permutations
 Finding Hamilton Paths and Circuits
 Hanoing
 Happy 8
 Hex Can't End in a Draw
 How to define a permutation
 Infinite Latin Square
 Jumping Isn't Everything
 Latin Squares
 Leibniz and Pascal Triangles
 Lifelike Automaton With Definable Rules
 Light Bulbs in a Circle
 Listing All Combinations
 Listing All Permutations
 Listing All Permutations: JohnsonTrotter Method
 Lucky 7
 Make Your Move, Kid!
 Marriage Problem, simulation
 Mazes
 Minumum Spanning Trees and Kruskal's Algorithm
 Monotone Number Sequences
 No Flipping without End
 Number sequences and Pigeonhole
 Party Acquaintances
 Permutations as a Product of Transpositions
 Product of Permutations
 Project Scheduling
 Property of the Powers of 2
 Random Walks
 Recombining Permutations
 Reflection Lemma
 Sam Loyd's Fifteen
 Scrub Tile Puzzle
 Shuttle Puzzle
 Simple Graphs
 Simple Graph Practice
 Sliders (slider games on torus, Moebius strip, projective plane, permutations, cycles)
 Straight Tromino on a Chessboard
 The Game of Life
 Tiling Rectangles with LTrominoes
 Tool for Constructing Orthogonal Latin Squares
 Tower of Hanoi
 Tower of Hanoi, the Hard Way
 Traveling Salesman Problem
 Two Men of Tibet
 Universal Coloring
 Y Can't End in a Draw
Combinatorial Games
 Date Game
 Dawson's Chess
 Dawson's Kayles
 Fraction Game
 Grundy's Game
 Hex 7
 Kayles
 Nim
 Nimble
 Northcott's game
 Odd Scoring
 One Pile
 Plainim
 Plainim Misère
 Scoring
 Scoring Misère
 Scoring Misère: Two Heaps Perfect Strategy
 Silver Dollar Game
 Silver Dollar Game With No Silver Dollar
 Sticks, Jess realization of Scoring Misère
 Sticky Problem
 Sticky Problem II
 Subtraction Game
 TacTix
 TakeAway Games
 Turning Turtles
 Wythoff's Nim
 Wythoff's Nim II
Puzzles & Games
 3Chain
 4×4×4 Tic Tac Toe (spacial visualization)
 A Game of Candy Squares
 Achieve the Equilibrium
 Bulls and Cows
 Caterpillar
 Changing Colors (combinatorics, invariance)
 Changing Colors II (invariance)
 Color Balls
 Computer Mastermind (logic and problem solving)
 Covering a Chessboard with a Hole with LTrominoes
 Eight Digits Puzzle
 Escape of the Clones
 Farmer and Wife To Catch Rooster and Hen (invariance, city block distance)
 Fif (multiple representations)
 Fill a Hexagonal Grid (space visualization)
 Fish Soup Game (multiple representations)
 Four Knights (multiple representations)
 Full House
 Getting a Scout out of Desert
 Goat, Cabbage and Wolf to move across a river (multiple representations)
 Hare and Hounds
 Jam (multiple representations)
 Looping Chips (multiple representations)
 Mastermind (logic and problem solving)
 Mastermind Variants (logic and problem solving)
 Mirror Maze
 Peg Solitaire (logic, problem solving, subproblems)
 Pick A Label (logic, problem solving)
 Reverse Solitaire (logic, problem solving, subproblems)
 Sending scouts into the desert (algebra, geometric series, problem solving)
 Seven Coins Puzzle (multiple representations)
 Shuttle Puzzle Practice I
 Shuttle Puzzle Practice II
 Skyscrapers Puzzle
 Solitaire on a Circle (iterations, invariance)
 Stones in Cups
 Super Nim
 Thugs and Bowmen
 Tiling a Triangulated Hexagon (a proof without words)
 Toads And Frogs in 2 dimensions (repeated application of same idea)
 Tromino Puzzle: Golomb's Theorem
 Tromino Puzzle: Deficient Squares
 Vector Addition and Subtraction
 Vowel Movement
 Wolf and Rabbit
Social Science
 Adams' Apportionment Method
 Banzhaf Power Index Calculator
 Fair Division: Method of Lone Divider
 Fair Division: Method of Markers
 Fair Division: Method of Sealed Bids
 Fair Division: Method of Sealed Bids II
 Fast Power Indices
 Five Methods of Apportionment
 Four Voting Methods
 Hamilton's Apportionment Method
 HuntingtonHill Apportionment Method
 Jefferson's Apportionment Method
 Method of Markers II
 ShapleyShubik Power Index Calculator
 Social Choice
 Webster's Apportionment Method
 Weighted Voting and Power Indices
Elementary Geometry
 5x5 Square Grid and 5 Circes
 Puzzle of 30 Grid Points and 5 Circles
 A Characterization of the Euler Line
 A Degenerate Case of Bottema's Configuration
 A Diameter As a Diagonal of Inscribed Quadrilateral
 A Double Meaning of an Arc's Midpoint
 A Family of Cyclic Quadrilaterals
 A Hinged Realization of a Plane Tessellation
 A Lemma of Equal Areas
 A Median Surprise
 A Multiplicative Identity of Areas in a Triangle
 A Property of a Median
 A Property of Equiangular Polygons
 A Property of Isogonal Lines
 A Property of Orthic Triangle
 A Property of Perspective Triangles
 A Three Pegs Question
 A Triangle With a 45 Degrees Angle in Square
 Acute Angle Bisectors in a Right Triangle
 AlNayrizi's Construction
 An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem
 An Extra Triple of Equilateral Triangles for Napoleon
 An Unexpected Rhombus
 Angle: An Illustrated Classification
 Angle Bisector
 Angle Bisectors in a Quadrilateral  Cyclic and Otherwise
 Angle Bisectors in Ellipse
 Angle Bisectors in Ellipse II
 Angle Bisectors On Circumcircle
 Angle Subtended by a Diameter
 Angles Inscribed in an Absent Circle
 Anticenter and Orthocenters
 Antiparallel
 Antiparallel via Three Reflections
 Archimedes' Law of the Lever
 Area of Median Triangle I
 Area of Median Triangle II
 Area of Parallelogram
 Area of the Intersection of Two Equal Rectangles
 Area of the Union of Two Squares
 Area of Triangle
 Areas between Two Parallelograms
 Areas in Square by Dissection
 Areas in Two Parallelograms and a Translation
 Arithmetic in Disguise
 Bespectacled Eyeballs Extension
 Between Major and Minor Circles
 Bottema with a Product of Rotations
 Bottema with Similar Triangles
 Bottema's Theorem  Proof Without Words
 Building a Bridge
 Building Bridges
 Carpet With a Hole
 Carpets in Triangle
 Center of Gravity and Motion of Material Points
 Chain of Four Intersecting Circles
 Characterization of Trapezoid
 Circle and Hyperbola as Lighthouse Curves II
 Circle Chains on Napoleon Triangles
 Circle Rotations and Fixed Points
 Circles and Semicircles in Rectangle
 Circle in a Square Inscribed in a Circle
 Circles with Equal Collinear Chords
 Circumcenter and Orthocenter
 Circumcenter on Angle Bisector
 Classification of Quadrilaterals
 Common Chord and a Tangent
 Concurrence in Arbelos
 Concurrence of Ten NinePoint Circles
 Concurrence on a Circle
 Concurrent and Parallel Lines in Parallelogram
 Concyclicity from Collinearity
 Concyclicity in Rectangle
 Compass Only Construction  A Chord Tangent to an Inner circle
 Concyclic Points from Midpoint of an Arc
 Concyclicity in Two Semicircles
 Conic from Parallel Chords
 Conic in Hexagon
 Conic Sections as Loci of Points
 Conics in Triangle
 Construct Triangle by Angle, Altitude and Median
 Construct Triangle by Angle, Median, and Circumradius
 Construction: Parallel Line through Point
 Construction of a Right Triangle
 Construction of an Oval  the Roman Way
 Construction of the Angle Bisector
 Construction of the Perpendicular Bisector
 Cross Points in a Polygon
 Cutting a Hole in Half
 Cyclic Quadrilateral, Angle Bisector And Isosceles Triangle
 Dancing Rectangles Model Auxetic Behavior
 Dancing Squares or a Hinged Plane Tessellation
 A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
 Diagonal Count
 Dissection of a Vase
 Dissection of Cyclic Quadrilateral
 Divide a Circle into N Parts of Equal Area
 E. W. Dijkstra's Proof of the Pythagorean Theorem
 Eight Equal Tangents
 Equal Areas in Equilateral Triangle
 Equal Areas in Regular 2ngons
 Equal Chords in Crossing Circles
 Equal Circles, Medial Triangle and Orthocenter
 Equidecomposition of a Rectangle and a Square
 Equidecomposition of Two Rectangles
 Equidecomposition of a Triangle and a Rectangle
 Equidecomposition of a Triangle and a Rectangle II
 Equidecomposition of Two Parallelograms
 Equilateral Triangle on a Closed Curve
 Equilateral Triangle on Parallel Lines
 Equilateral Triangle on Parallel Lines II
 Equilateral Triangle on Three Lines
 Equilateral Triangles formed by Circumcenters
 Equilateral Triangle in Equilateral Triangle
 Equilic Quadrilateral I
 Equilic Quadrilateral I, A Variation
 Equilic Quadrilateral II
 Euclid's Segment Division
 Existence of the Circumcenter
 Existence of the Euler Line: An Elementary Proof
 Existence of the Orthocenter
 Extras in Bottema's Configuration
 EyetoEye Theorem I
 EyetoEye Theorem II
 Fibonacci Bamboozlement
 Famous Theorem
 Finding Circle Center By Ruler Alone
 Five Squares in Complex Numbers
 Fixed Point in Isosceles and Equilateral Triangles
 Fixed Point on Perpendicular Bisector
 Focal Definition of Ellipse
 Focus and Directrix of Ellipse
 Folding Square in a Line through the Center
 Four Circles and a Line
 Four Hinged Squares
 From Arbitrary Hexagon to Regular One
 From Perpendicular Center Lines to Concyclic Points
 From Straight Line to Regular Hexagon
 From Arbitrary Pentagon to Regular One
 Garcia's Two Circles Lemmas
 Geoboard
 Half Turn, Reflection in Point
 Harmonic Mean Sangaku
 Hinged Greek Cross Tessellation
 Hippocrates' Squaring of a Lune
 Homothety in Three Tangent Circles
 Homothety in Three Tangent Circles II
 How to Construct Tangents from a Point to a Circle
 Hunting Right Angles
 Huygens' proof of the Pythagorean Theorem
 In the Spirit of Thebault I
 Inscribed and Central Angles in a Circle
 Inscribed Angles
 Inscriptible Quadrilateral: An Illustration
 Integer Triangles with Two Angles in the Ratio 1 : 2
 Intersecting Chords Theorem  a Visual Proof
 Intersecting Chords Theorem  Hubert Shutrick's PWW
 Intersections of a Circle with the Four Quadrants
 Invariance in Orthodiagonal Quadrilaterals
 Is X a Midpoint of a Chord
 Isogonal Lemma
 Isogonal Concurrencies
 Isoperimetric Theorem For Quadrilaterals
 Isoperimetric Theorem For Quadrilaterals II
 Isosceles Configuration in Triangle
 Join Circles by Given Segment
 Joined Common Chords of Napoleon's Circumcircles
 Law of Cosines  Another PWW
 Lean Napoleon's Triangles
 Line Not Through a Center of Similarity
 Line Through a Center of Similarity
 Looking Back at Bottema
 Maimed Cake
 Mathematics in Pizzeria
 Midline in Trapezoid
 Midline in Triangle
 Miguel's Area of Square
 Miguel Ochoa's Chords And Tangents
 Miguel Ochoa's van Schooten is a Slanted Viviani
 Miguel Ochoa's van Schooten Like Theorem
 Miguel Ochoa's van Schooten Like Theorem II
 Miguel Ochoa's van Schooten Like Theorem III
 Miquel Circumcenters
 Napoleon's Hexagon
 Napoleon's Theorem: A third proof with complex numbers
 Napoleon's Theorem via Two Rotations
 Napoleon on Hinges
 Napoleon on Hinges in GeoGebra
 Neuberg Cubic
 Nine Point Circle: an Elementary Proof
 Octagon in Parallelogram
 Octagon in Parallelogram: Overlapping Regions
 Olof Hanner's Jigsaw Puzzle
 Orthic Semiperimeter
 Orthocenter and Circumcenter Are Isogonal Conjugate
 Orthocenters of Two Triangles Sharing Circumcenter and Base
 Orthodiagonal and Cyclic Quadrilaterals
 Orthogonality in Two Squares
 Pappus' Generalization of Euclid I.47
 Parabolic Reciprocity
 Parallel Chords II
 Parallel Chords in Crossing Circles
 Parallel Lines in a Quadrilateral
 Parallel Lines in a Quadrilateral II
 Parallelogram and Four Equilateral Triangles
 Parallelograms among Quadrilaterals
 Parallelogram and Ellipses
 Parallelogram in Four Circles
 Parallelogram in Parallelogram And in Projection
 Parallelogram with Side Lines through Fixed Points
 Pascal: Necessary and Sufficient
 Pedal Similarity
 Pedal Triangles of Inverse Points
 Perimeters of Convex Polygons, One within the Other
 PLL: Apollonius' Problem with Two Lines and a Point
 Polygon Metamorphosis
 Pompeiu's Theorem
 Praying Eyes Theorem
 Problem 1 from the EGMO2017
 Proclivity of Conjoint Archimedean Twins for Proliferation
 Properties of Flank Triangles  Second Proof by Symmetric Rotation
 Properties of Flank Triangles  Proof with an Asymetric Rotation
 Property of Angle Bisectors
 Property of Angle Bisectors III
 Property of Internal Angle Bisector  Hubert Shutrick's PWW
 Property of Rhombi
 Property of Right Isosceles Triangle
 Property of Semicircles
 Pure Angle Chasing
 Pure Angle Chasing II
 Pure Angle Chasing III
 Pythagorean Theorem By Dissection
 Pythagorean Theorem By Dissection II
 Pythagorean Theorem by Hinged Dissection
 Pythagorean Theorem by Hinged Dissection I
 Pythagorean Theorem by Hinged Dissection II
 Pythagorean Theorem: J. Adams' Proof by Dissection
 Pythagorean Theorem: Versluys' Proof by Dissection
 Pythagoras' Theorem By Sheer Shearing
 Pythagoras' Theorem By Similarity
 Pythagoras' Theorem Solely By Sliding
 Pythagorean Theorem via an Isosceles Triangle
 Pythagorean Segment Division
 Quadrilateral With Equal Opposite Sides And Angles
 Rectangles among Parallelograms
 Rectangular Areas in Circle
 Reflection in the Circumcenter
 Regular Hexagon and Triangle Joined at Vertex
 Reim's Similar Coins I
 Reim's Similar Coins II
 Reim's Similar Coins III
 Reim's Similar Coins IV
 Rhombus in a Cyclic Quadrilateral
 Right Triangles on Sides of a Square
 Rhombus in Circles
 Sangaku with Angle between a Tangent and a Chord
 Secant Angles
 Secant Angles II
 Secant, Normal, Tangent
 Segment Trisection II
 Segment Trisection Induced by Parallels to Medians
 Semiregular Tessellation on Hinges A
 Semiregular Tessellation on Hinges B
 Semiregular Tessellation on Hinges C
 Seven Concyclic Points
 Shearing a Polygon into a Triangle of Equal Area
 Shifted by the Inradius Tangent Collinearities
 Similar Triangles in Crossing Circles
 Simple Property of Circle Through the Incenter
 Simple Quadrilaterals Tessellate the Plane
 Shearing and Translation in Pythagorean Pants
 Some Properties of Common Tangents
 Spiral Similarity Leads to Equilateral Triangle
 Square, Semicircle, and Other Curves
 Square From Nowhere
 Square in a Circle Inscribed in a Square
 Square in Chair
 Squaring a Rectangle
 Sum of Two Squares
 Symmetries in Triangle II
 Symmetry in an Asymmetric Configuration
 Tangent Circles, Similarity and Equal Segments
 Tangent Curiosity in Equilateral Triangle
 Thales' Theorem
 Thales on Angle Bisectors
 The Eutrigon Theorem
 The Shape of the Colosseum
 Three Collinear and Four Concyclic Points
 Three Common Chords in Three Concurrent Circles
 Three lines at 60 degres and Circumcenters
 Three Touching Circles
 Translated Triangles
 Translation Transform
 Triangle Areas in a Parallelogram
 Triangle Areas in a Parallelogram II
 Triangle Classification
 Trigonometric Functions
 Triangular Areas in Circle
 Twin Segments in Four Semicircles
 Two Altitudes, One Midpoint
 Two Color Coloring of the Plane
 Two Common Chords in Three Concurrent Circles
 Two Diameters and Longest Common Chord
 Two Equilateral Triangles
 Two Intersecting Circles
 Two Parallels in a Triangle and One More
 Two Pascals Merge into One
 Two Properties of Flank Triangles  First Proof by Symmetric Rotation
 Two Properties of Flank Triangles  A Proof with Complex Numbers
 Two Properties of Flank Triangles  and a Third One
 Two Regular Pentagons Joined at Vertex
 Two Symmetric Triangles Are Directly Equidecomposable
 Two Symmetric Triangles Are Directly Equidecomposable II
 Two Symmetric Triangles Are Directly Equidecomposable III
 Two Symmetric Triangles Are Directly Equidecomposable IV
 Two Touching Circles
 Van Schooten's and Pompeiu's Theorems
 Viviani's Theorem
 Viviani's Theorem II
 Viviani in Isosceles Triangle
 Yet Another Generalization of Bottema's Theorem
Stereometry
 Angles in a Cube I
 Angles in a Cube II
 Angles in a Cube III
 Bimedians in a Regular Tetrahedron
 Cut the Cone
 Cut the Cube
 Cut the Cylinder
 Knots on a Torus
 Möbius' Strip
 Multifaceted Cork
 One Sheet Hyperboloid
 Platonic Solids  Regular Polyhedra
 Other Solids


Geometric Fallacies
 A Circle With Two Centers
 All Triangles Are Isosceles
 Curry's Paradox
 Delian Problem Solved
 A Faulty Dissection
 Hooper's Paradox
 Is Every Trapezoid Parallelogram?
 John Sharp's Paradox
 Langman's Paradox
 Popping A Square
 Rouse Ball's Fallacy
 Sam Loyd's Son's Dissection
 Two Perpendiculars From a Point to a Line
Outline (FillItIn) Mathematics

A bunch of fillitin outline problems and solutions
Geometry
 120° Breeds 90°
 2NWing Butterfly Theorem
 3 Isosceles Trapezoids
 5Star and A Circle
 6 to 9 Point Circle
 60° Breeds 90°
 9point Circle as a Locus of Concurrency
 9 Point Center on Angle Bisector
 A Chain of Touching Circles in a Polygon (à la Quang Tuan)
 A Chain of Touching Circles in a Polygon (à la Evelyn)
 A Circle Related to Incenter and Circumcenter
 A Circle Rolling in an Equilateral Triangle
 A Conic in a Triangle
 A Generalization of Simson Line
 A Geometry Problem from 2012 IMO  Problem 5
 A Line in Triangle Through the Circumcenter
 A Parallelogram in Triangle
 A Problem in Three Pentagons
 A Sangaku FollowUp on an Archimedes' Lemma
 A Sangaku: Two Unrelated Circles
 A Sangaku with an Egyptian Attachment
 A Sangaku with Many Circles and Some
 A 60° Rhombus in a 60° Isosceles Trapezoid
 A Stronger Triangle Inequality
 A Triangle of Antreas Hatzipolakis
 Abutting Equilateral Triangles on a Line I
 Adams' Circle
 Addition of RadiusVectors
 Altitudes and the Power of a Point
 An Euclidean Construction with Inversion
 An Isoperimetric Theorem
 An Unexpected Pair of Similar Triangles Which Are Equal
 Angle Bisector in Equilateral Trapezoid
 Angle Bisector in Parallelogram
 Angle Bisector in Touching Circles
 Angle Bisectors and Perpendiculars in a Quadrilateral
 Angle Bisectors in a Quadrilateral
 Angle Preservation Property
 Angle Trisection by Archimedes
 Angle Trisection by Hippocrates
 Angle Trisection by Paul Vjecsner
 Angle Trisectors on Circumcircle
 Angles in Triangle Add to 180°
 Another Concurrence on the 9Point Circle
 Another Geometry Problem from 2012 IMO  Problem 1
 Another Equilateral Triangle in Napoleon's Configuration
 Another Pair of Twins in an Arbelos
 Another Property of the 9Point Circle
 Another Seven Circles Theorem
 Anticomplementary Triangle Surprise
 Antiparallel and Circumradius
 Apollonian Circle for Two Lines and a Circle
 Apollonian Circles Theorem
 Apollonius Problem
 Application of Ceva's Theorem
 Archimedean Siblings out of Wedlock, i.e., Arbelos
 Archimedes' Quadruplets
 Archimedes Triangle and Squaring of Parabola
 Archimedes' Twin Circles and a Brother
 Area of an Arbelos
 Areas and Centroid in a Triangle
 Areas in Three Squares
 Areas in Triangle
 Areas in Triangle II
 Arithmetic Mean Sangaku
 Arithmetic Mean Sangaku II
 Around the Incircle
 Around the Incircle in a Polygon
 Asian Pacific Kite
 Asymmetric Propeller I
 Asymmetric Propeller II
 Asymmetric Propeller III
 Asymmetric Propeller by Plane Tiling
 Barycentric Coordinates: a Tool
 Batman's Problem
 Bevan's Point and Theorem
 Bisectal Circle
 BMO 2014, Problem 3
 Book of Lemmas: Proposition 1
 Book of Lemmas: Proposition 2
 Book of Lemmas: Proposition 3
 Book of Lemmas: Proposition 5
 Book of Lemmas: Proposition 6
 Book of Lemmas: Proposition 9
 Book of Lemmas: Proposition 10
 Book of Lemmas: Proposition 12
 Book of Lemmas: Proposition 13
 Böttcher's Proof of the Pythagorean Theorem
 Böttcher's Proof Variant
 Bottema Shatters Japan's Seclusion
 Bottema's Theorem
 Bottles in a Slanted Rack
 Bottles in a Wine Rack
 Bottles in a Wine Rack: Proofs and Generalizations
 Brahmagupta's Theorem
 Brianchon in Ellipse
 Brianchon's Theorem
 Bride's Chair
 Broken Chord Theorem
 Broken Chord Theorem by Paper Folding
 Broken Chord Theorem: G. Patruno's proof
 Broken Chord Theorem: S. Anderson's proof
 Butterflies in a Pencil of Conics
 Butterfly in Ellipse
 Butterfly in Parabola
 Butterfly Problem
 Cabart's Collinearity
 Cantor's Theorem
 Cardioid As Envelope of Circles
 Cardioid As Epicycloid
 Carnot's Theorem
 Carpets in a Parallelogram
 Carpets in a Quadrilateral
 Carpets in a Quadrilateral II
 Carpets Theorem
 Carpets Theorem With Parallelograms
 Casey's Theorem
 Cassini's Ovals and Geometric Optimization
 CCC: Apollonius' Problem with Three Intersecting Circles
 CCC: Gergonnes's Solution to Apollonius' Problem
 Center Circles and Their Chains
 Centroids in a Polygon
 Ceva and Menelaus Theorems for Angle Bisectors
 Ceva's Theorem
 Ceva's Theorem (PWW)
 Cevian Cradle
 Cevian Cradle II
 Cevian Nest
 Cevian Parallelogram
 Cevian Triangle
 Chain of Circles on a Chord
 Chain of Inscribed Circles
 Chain of Six Intersecting Circles
 Chasing Angles in Pascal's Hexagon
 Chasles' Theorem, a Proof
 Cherchez le quadrilatère cyclique
 Cherchez le quadrilatère cyclique II
 Chords, Concurrency and Orthic Triangle
 Chords in Cardioid
 Circle and Hyperbola as Lighthouse Curves
 Circle Centers on Radical Axes
 Circle Chains and Inscribed Angles
 Circle Concurrence on Circumcircle/a>
 Circle Concurrency and Spiral Similarity
 Circle Crosses Triangle
 Circle Division by Euclid and Leonardo
 Circle Inscribed in a Circular Segment
 Circle of Similitude
 Circle through the Circumcenter
 Circle through the Incenter
 Circle through the Incenter And Antiparallels
 CircleStacking Theorem
 Circle, Isosceles Triangle and Fixed Point
 Circles And Parallels
 Circles and Reflections in a Triangle
 Circles Cover a Quadrilateral
 Circles in a Regular Polygon
 Circles in Morley's Triangles
 Circles Lined on the Legs of a Right Triangle
 Circles On Cevians
 Circles Tangent to Circumcircle
 Circles Tangent to Medians at the Centroid
 Circles through the Orthocenter
 Circles with Tangent Diameters
 Circumcevian Triangle
 Cleavance Center
 Cleaver
 Clifford Chain
 Clifford's Lemma
 Coaxal Circles on Perpendicular Bisector
 Coaxal Circles Theorem
 Colburn's Proof of the Pythagorean Theorem
 Collinear Intersections and Products of Ratios
 Collinearity from the IMO 2013 (Problem 4)
 Collinearity via Concyclicity
 Collinearity with the Orthocenter
 Common Centroids Lead to Equilateral Triangle
 Common Tangents to Two Circles I
 Common Tangents to Two Circles II
 Complete Quadrilateral
 Concurrence in Equilateral Triangle
 Concurrence in Right Triangle
 Concurrence Not from School Geometry
 Concurrence of the Altitudes As Seen from 3D
 Concurrency in Analytic Geometry
 Concurrency in Arbelos
 Concurrency in the Intouch Triangle
 Concurrent Cevians and a Conic through Their Feet
 Concyclic Circumcenters: A Dynamic View
 Concyclic Points in a Triangle
 Concyclic Incenters in Bicentric Quadrilateral
 Concyclic Points at the IMO 2014
 Concyclic Points in Bride's Chair
 Concyclic Points in Equilateral Bumps
 Concyclic Points in Inscriptible Quadrilateral
 Concyclic Points of Two Ellipses with Orthogonal Axes
 Conic in Mixtilinear Incircles
 Conjugate Diameters in Ellipse
 Construct Triangle by Angle Bisector, Altitude, and Side
 Construction and Properties of Mixtilinear Incircles
 Construction and Properties of Mixtilinear Incircles 2
 Construction of Conics from Pascal's Theorem
 Construction of Thebault Circles
 Consecutive Isosceles Decomposition
 Conway's Circle
 Copernicus' Theorem
 CrossedLines Construction of Shapes of Constant Width
 Cutting Triangle in Two by a Line through a Point
 Cyclic Hexagon
 Cyclic Incenters
 Cyclic Quadrilateral, Concurrent Circles and Collinear Points
 Cycloids
 Dao's Archimedean Twins
 Dao's Variant of Thébault's First Problem
 David Richeson's Extention of an Old Japanses Theorem
 Diameters and Chords
 Directly Similar Figures
 Desargues' Hexagon
 Desargues' Theorem
 Discovery of Duane DeTemple
 Dissection of Triangle into Rhombus
 Dissection of Triangle into Rhombus by Hubert Shutrick
 Divide Triangle by Lines Parallel to Base
 Dividing a Segment into N parts: Besteman's Construction
 Dividing a Segment into N parts: Besteman's Construction II
 Dividing a Segment into N parts: GLaD Construction
 Dividing a Segment into N parts: SaRD Construction
 Dividing Evenly a Quadrilateral
 Dividing Evenly a Quadrilateral II
 Double Generation Theorem
 Douglas' Theorem (complex variables, linear algebra)
 DrozFarny Circles
 DrozFarny Line Theorem
 DrozFarny Theorem  an Analytic Solution
 Dual of Maxwell's Theorem
 Dual to Pappus' Theorem
 Dynamic Construction of Ellipse and Other Curves
 Dynamic Software as Serendipity Enhancement
 Easy Construction of Bicentric Quadrilateral
 Easy Construction of Bicentric Quadrilateral II
 Eight Point Circle
 Ellipse, how to draw
 Ellipse Between Two Circles
 Ellipse in Arbelos
 Ellipse Touching Sides of Triangle at Midpoints
 >Emil Stoyanov's New Year's Problem
 Equal Angles in Two Circles
 Equal Angles in Two Circles II
 Equal Areas In Parallelogram à la Pythagoras
 Equal Incircles Theorem
 Equichordal Curves
 Equilateral and 345 Triangles
 Equilateral Triangle, Straight Line and Tangent Circles
 Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
 Equilateral Triangles on Segments of Equilic Quadrilateral
 Equilateral Triangles On Sides of a Parallelogram
 Equilateral Triangles On Sides of a Parallelogram II
 Equilateral Triangles on Sides of a Parallelogram III
 Equilateral Triangles on Sides of a Quadrilateral
 Escher's Theorem
 Euler Line Cuts Off Equilateral Triangle
 Euclid I.43
 Euclid I.43 Extended
 Euclidean Construction of Center of Ellipse
 Euclidean Construction of Tangent to Ellipse
 Excircles in Trapezoid
 Exeter Point
 Extouch Triangle in Poncelet Porism
 Extra Feature of Van Aubel Configuration
 Extremal Distance Ratio
 Eyeball Theorem
 Eyeballing a Ball
 Eyeballs Projected
 External Angle Bisector
 Fagnano's problem
 Fagnano's problem II
 Fagnano's Problem, Third Solution
 Fagnano's Problem, Fourth Solution
 Fagnano's Problem in Reverse
 Fermat's Hexagon
 Fermat point and 9point Centers
 Fermat's Point
 Fermat Points and Concurrent Euler Lines I
 Fermat Points and Concurrent Euler Lines II
 Feuerbach's theorem
 Feuerbach's Theorem: A Proof
 Final Chapter of the Asymmetric Propeller Story
 Find a Common Chord of Given Length
 Find the Center of a Given Circle
 First Vecten Point
 Five Cardioids
 Five Concyclic Points
 Five Incircles in a Square
 Five Incircles Theorem
 Fixed Point of Circles Orthogonal to the Given One
 Fixed point on a rod rotating around a triangle
 Focal Properties of Parabola
 Focus on the Eyeball Theorem
 Foot of Altitude and Minimum Distance
 For Equality Choose Angle Bisector
 Four 9Point Circles in a Quadrilateral
 Four Centroids and Parallels
 Four Circles In a Triangle
 Four Concurrent Lines in a Cyclic Quadrilateral
 Four Crossing Circles
 Four Incident Circles
 Four Incircles in Equilateral Triangle
 Four Isosceles Triangles Hinged at Vertices
 Four Pedal Circles
 Four Touching Circles
 Four Touching Circles II
 Four Touching Circles III
 The Four Touching Circles Problem IV
 Four Triangles, One Circle
 Four Turtles (or Bugs, or Dogs, etc.)
 François Viète's Reduction of CCC to PCC
 Friendly Kiepert's Perspectors
 From Foci to a Tangent in Ellipse
 Fundamental Theorem of 3Bar Motion
 Further properties of Van Aubel Configuration
 Fuss' Theorem
 Garcia's Archimedean Quadruplets
 GarciaFeuerbach Collinearity
 Generalizing a Romanian Olympiad Problem
 Geometric Optimization from the Asian Pacific Mathematical Olympiad
 Gergonne and Medial Triangles Are Orthologic
 Gergonne and Soddy Lines Are Perpendicular
 Gergonne in Ellipse
 Glide Reflection
 Golden Ratio in an Irregular Pentagon, Construction I
 Golden Ratio in an Irregular Pentagon, Construction II
 Golomb's inductive proof of a tromino theorem
 Gothic Arc
 Grebe: from Ladies' Diary to Carroll's Pillow
 Grebe's Theorem
 Greg Markowsky's Problem for Parabola
 Griffiths Points in Cyclic Quadrilateral
 Harmonic Ratio
 Hart Circle
 Hart's Inversor
 Haruki's Lemma
 Heron's Formula: a Proof
 Heron's Problem
 Heron in Trapezoid
 Hexagon Parallel to Medial Triangle
 Hexagon Parallel to Orthic Triangle
 Hidden Orthogonality
 Hinged Squares
 Hinged Squares II
 Hjelmslev Theorem
 Hjelmslev's Theorem on Canvas
 Hofstadter Triangles and Points
 Homologous Lines under Three Spiral Similarities
 Homothety
 Homothety between In and Excircles
 Homothety in Equilateral Triangles
 How to Construct a Radical Axis
 How to Construct Orthogonal Circle
 Hypocycloid Families
 Ian McGee's Observation
 Illustration of Pascal's Theorem
 Incenters in Cyclic Quadrilateral
 Incidence in Feuerbach's Theorem
 Incident Center Circles and Cardioid
 Incircles, Circumcircles and Inscribed Angles
 Inscriptible and Exscriptible Quadrilaterals
 Inscriptible Quadrilateral of Triangle Incenters
 Internal Tangents to Three Circles
 Intouch Triangle in Poncelet Porism
 Invariant Area Sweep Implies Pythagorean Theorem
 Inversion: Reflection in a Circle
 Inversion in the Incircle
 Inversion Tool
 Inversion with Negative Power
 Invert Two Circles Into Equal Ones
 Isogonal and Isotomic Conjugalities
 Isogonal Concurrency
 Isogonal image of the circumcircle
 Isosceles on the Sides of a Triangle
 Isosceles Trapezoid In Cyclic Quadrilateral
 Isotomy and Isogonality HandinHand
 Iterations in Geometry, an example
 Iterations in Geometry, Generalization
 Iterations on a Circle Through Three Points
 Iterations on Euler Lines
 Johnson Circles
 Joseph Keech in Bride's Chair
 Kiepert's Centroid
 Kiepert's Theorem
 Kiepert's Triangles Graduate to Ears of Arbitrary Shape
 Kürschak's Tile and Theorem
 La Hire's Theorem
 La Hire's Theorem, a Variant
 La Hire's Theorem in Ellipse
 Law of Cosines
 Law of Cosines: PWW
 Law of Cosines: An Unfolded Variant
 Law of Cosines by Ancient Sliding
 Lighthouse Curves
 Line, Circle, and Fixed Points
 LCC: Apollonius' Problem with Two Circles and a Line
 LLC: Apollonius' Problem with Two Lines and a Circle
 Locus of Points in a Given Ratio to Two Points
 MacLaurin's Construction of Conics
 Malfatti's Problem
 Malfatti's Problem, Hart's Solution
 Maximum Perimeter Property of the Incircle
 MathPro Logo
 Maxwell's Theorem
 Median in Touching Circles
 Medians in a Quadrilateral
 Menelaus and Ceva Theorems
 Menelaus From Ceva
 Midline in Quadrilateral
 Midline in Similar Triangles
 Midpoint Reciprocity in Napoleon's Configuration
 Midpoints and Orthogonality in Isosceles Triangles
 Midpoints from Gergonne Triangle
 Midpoints of the Lines Joining In and Excenters
 Miguel's Tangents: Modern Day Sangaku
 Minkowski's addition of convex shapes
 Miquel's Point
 Miquel's Point of a 4line Via Spiral Similarity
 Miquel's Theorem for Circles
 Mirror on the Wall
 Mirror property of the altitudes
 Mirror Property of Altitudes via Pascal's Hexagram
 Mixtilinear Circles and Concurrence
 Monge via Desargues
 Monge via Desargues II
 More Bottles in a Wine Rack
 Morley Constellation
 Morley's Equilaterals, Spiridon A. Kuruklis' proof
 Morley's Theorem
 Morley's Theorem, Cardioids and Such
 Morley's Theorem, G. Zsolt Kiss' proof
 Morley's Theorem: Taylor and Marr's Proof
 Nagel Line
 Nagel point
 Nagel Point of Medial Triangle
 Nagel's Theorem
 NapoleonBarlotti Theorem
 Napoleon Theorem by Plane Tessellation
 Napoleon's Relatives
 Napoleon's Theorem by Transformation
 Neuberg Sangaku
 Newly Born Pair of Siblings to Archimedes' Twins
 Newton's Construction of Conics
 Newton's Theorem
 Nine Point Circle
 No Equilateral Triangles, Please
 Nobbs' Points, Gergonne Line
 Nonclassic Way To Draw Cardioid
 NoPedal Collinearity
 Not So Hidden Homotheties
 Novel Concurrency of Perpendicular Bisectors
 On Bottema's Shoulders
 On Bottema's Shoulders II
 On Bottema's Shoulders with a Ladder
 One More Seven Circles Theorem
 One Sided Surface in 4D
 Optical Property of Ellipse
 Optimal Quadrilateral Inscribed in Square, PWW
 Optimization Problem in Acute Angle
 Original Taylor and Marr's Proof
 Orthogonal Lines, Midpoints, and Collinearity
 Orthocenter and Three Equal Circles
 Orthocentric System From Rectangle
 Orthogonality in Isogonal Conjugacy
 Orthogonality in Isosceles Triangles
 Orthologic Triangles in a Quadrilateral
 Orthopole
 Pairs of Homologous Lines under Spiral Similarities
 Pairs of Incircles in a Quadrilateral
 Pantograph
 Pappus' Theorem
 Parabola As Envelope of Straight Lines
 Parabola As Envelope II
 Parabolas Related to the Orthic Triangle
 Parabolic Mirror, Illustration
 Parabolic Mirror, Theory
 Paragon
 Parahexagon
 Parallel Chords
 Parallel Chords in Ellipse
 Parallel Lines and Similar Triangles
 Parallel Lines in a Cyclic Quadrilateral
 Parallel Lines in Rhombus
 Parallel Lines in a Triangle
 Parallelogram and Similar Triangles
 Parallelogram Iterations
 Parallelogram Law
 Parallelogram Law: A PWW
 Parallelogram Law: A Tessellation
 Pascal in a Cyclic Quadrilateral
 Pascal Lines: Steiner and Kirkman Theorems
 Pascal Lines: Steiner and Kirkman Theorems II
 Pascal's Theorem
 Peaucellier Linkage
 Pedal Collinearities
 Pedal Parallelogram
 Pedal Polygons
 Pedal Triangle and Isogonal Conjugacy
 Pedoe's Theorem
 Pentagon in a Semicircle
 Perigal's Proof of the Pythagorean Proposition
 Perimeters of Parallelogram And Rhombus
 Perpendicular Bisectors in an Inscriptible Quadrilateral
 Perpendicular Bisectors in an Inscriptible Quadrilateral II
 Pinning Butterfly on Radical Axes
 Pivot Theorem
 Point common to two similar rectangles
 Polar Circle
 Pole and Polar with respect to a Triangle
 Poles and Polars
 Poles, Polars and Orthogonal Circles
 Poles, Polars and Quadrilaterals
 Polygon from the midpoints of its sides
 Polygonal Ears and Triangulation Duals
 Polygons Formed by Perpendicular Bisectors
 Poncelet's Porism
 Poncelet's Porism in Ellipses
 Poncelet's Theorem
 PCC: Apollonius' Problem with Two Circles and a Point
 PLC: Apollonius Problem with a point, a line, and a circle
 PPC: Apollonius' Problem with Two Points and a Circle
 PPL: An Elementary Solution
 PPL: Apollonius' Problem with Two Points and a Line
 Problem 4, 1975 USA Math Olympiad and the Radical Axis
 Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
 Problem 4, 1975 USA Math Olympiad: Normals and Tangents
 Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch
 Problem 4 from the IMO 2013  Synthetic Solution
 Problem in an Equilateral Triangle
 Problem in an Equilateral Triangle II
 Problem In an Isosceles Right Triangle
 Problem in Three Squares
 Problem of Equal Steps
 Problem of Equal Steps II
 Problem of polygon construction
 Problem with Equilateral Triangles
 Product of Rotations
 Projections of Convex Quadrilateral
 Projections of Isogonal Conjugate are Concyclic
 Projections on Internal and External Angle Bisectors
 Projective Collinearity in a Quadrilateral
 Projective Generalization of Maxwell's Theorem
 Projective Proof of Pascal's Theorem
 Proof Without Words for the Pythagorean Theorem
 Properties of the Circle of Similitude
 Property of 6Parpolygon
 Property of Angle Bisectors II
 Property of Circumscribed Quadrilaterals
 Property of Points Where In and Excircles Touch a Triangle
 Property of Right Trapezoids
 Property of the Line IO
 Property of the Line IO II
 Property of the Line IO: a proof from the Book
 Property of Thébault Circles
 Proportions with Two Orthocenters
 Ptolemy on Hinges
 Pythagoras and Vecten Break Japan's Isolation
 Pythagoras' from Bottema's
 Pythagorean Condition in An Isosceles Right Triangle
 Pythagorean Theorem: Subtle Dangers of Visual Proof
 Pythagorean Theorem By Euclid
 Pythagorean Theorem By Plane Tessellation
 Pythagorean Theorem By Hexagonal Tessellation
 Pythagorean Theorem By Rearrangement
 Pythagorean Theorem By Shearing and Rotations I
 Pythagorean Theorem By Shearing and Rotations II
 Pythagorean Theorem in an Orthodiagonal Quadrilateral
 Quadrilaterals Formed by Perpendicular Bisectors
 Quadrilateral with 3 Equal Sides
 Quest for Paragon
 Radical Axis and Center, an Application
 Radical Axis of Circles Inscribed in a Circular Segment
 Radical Center
 Radius and Construction of a Mixtilinear Circle
 Radius of a Circle by Paper Folding
 Rectangle in Arbelos
 Rectangle Inscribed in Triangle
 Rectified, Halved, Sheared, Eyeballs Still Surprise
 Reflection in a Triangle Tangent to Parabola
 Reflection In Line
 Reflections in Ellipse
 Reflections of a Line Through the Orthocenter
 Reflections of a Point on the Circumcircle
 Reflections of the Orthocenter
 Regular Polygons in a Triangular Grid
 Remarkable Line in a Quadrilateral
 Remarkable Line in Cyclic Quadrilateral
 Rendezvous
 Surprise: Right Angle in Circle
 Right Isosceles Triangles on Sides of a Quadrilateral
 Rotation Transform
 Rotations in Disguise
 Rusty Compass Construction of Equilateral Triangle
 Salinon: from Archimedes Book of Lemmas
 Sanchez's Viviani's Area Analogue
 Sangaku Iterations, Is it Wasan?
 Sangakus with a Mixtilinear Circle
 Sangaku with Three Mixtilinear Circles
 Seven Circles Theorem
 Squares in Arbelos
 Secant, Tangents and Orthogonality
 SeeSaw Lemma
 Segment Trisection
 Seven Equal Circles
 Shape of Constant Width
 Shearing Butterflies in Quadrilaterals
 Siamese Triangles
 Siamese Triangles II
 Similar Triangles on Sides of a Quadrilateral
 Similar Triangles on Sides and Diagonals of a Quadrilateral
 Simson Line
 Simson Line From Isogonal Perspective
 Simson Line in Disguise
 Simson Lines Of Diametrically Opposite Points
 Simsons and 9Point Circles in Cyclic Quadrilateral
 Simultaneous Diameters in Concurrent Circles
 sin(N)
 Six Circles with Concurrent Pairwise Radical Axes
 Six Circles Theorem (Bui Quang Tuan)
 Six Circles Theorem (Elkies)
 Six Circles Theorem (Evelyn)
 Six Concyclic Circumcenters in Fermat's Configuration
 Six Concyclic Points
 Six Concyclic Points II
 Six Concyclic Points via Antipedal Triangle
 Six Incircles in an Equilateral Triangle
 Six Point Circle
 SixPoints, ThreeLines
 Soddy Circles and David Eppstein's Centers
 Some Properties of Napoleon's Configuration
 Spectral Decomposition of a Planar Quadrilateral: an Illustration
 Sperner's Lemma
 Spira Mirabilis  Wonderful Spiral
 Spiral Similarity
 Spiral Similarity and CollinearitySpiral Similarity and Collinearity
 Square from Four Points, One on Each Side
 Square in a Right Triangle
 Square Inscribed in a Triangle
 Square Inscribed in a Triangle II
 Square Inscribed In a Triangle III
 Square out of a Quadrilateral
 Square, Similarity and Slopes
 Squares in Circles
 Squares on Sides of a Parallelogram
 Squares on Sides of a Quadrilateral
 Squares on Sides of a Quadrilateral II
 Squinting Eyes Theorem
 Star Construction of Shapes of Constant Width
 Steiner's Circumellipse in Barycentric Coordinates
 Steiner's Chain
 Steiner's porism
 Straight Edge Construction of Polar
 Surprising 60° in Equilateral Triangle
 Symmedian and Antiparallel
 Symmedian and 2 Antiparallels
 Symmedian and the Simson
 Symmedian and the Tangents
 Symmedian in a Right Triangle
 Symmedian in Right Triangle II
 Symmedian via Parallel Transversal and Two Circles
 Tangency Point of Two Circles
 Tangent and Secant
 Tangent as a Radical Axis
 Tangent Circles and an Isosceles Triangle
Tangent Circles and an Isosceles Triangle II
 Tangent Circles in a Parallelogram
 Tangent in Concurrency
 Tangent Lines and Circles in Convex Quadrilateral
 Tangents and Diagonals in Cyclic Quadrilateral
 Tangents, Perpendiculars and Geometric Mean
 Terquem's Theorem
 Tesseract
 The Midpoint of an Angle Bisector  a Problem by A. A. Zaslavsky
 The Plain Butterfly Theorem
 The Shortest Crease
 Thébault's Problem I
 Thébault's Problem II
 Thébault's Problem III
 Thébault's Problem IV
 Thébault's Problem III, Proof
 Theorem of Complete Quadrilateral
 Theorem of Three Tangents to a Conic
 Theorems of Ceva and Menelaus, an Illustrated Generalization
 Three Circles through the Incenter
 Three Concurrent Circles
 Three Congruent Circles by Reflection
 Three Congruent Circles by Reflection II
 Three Congruent Circles by Reflection III
 Three Incircles In a Right Triangle
 Three Incircles in a Triangle
 Three Isosceles Triangles
 Three Orthogonal Circles Through Three Given Points
 Three pairs of Parallel Lines and a Conic
 Three Parabola Tangents
 Three Parabolas with Common Directrix
 Three Parallels in a Triangle
 Three Points Casey's Theorem
 Three Points on a Parabola
 Three Roads, Three Travelers
 Three Similar Polygons
 Three Similar Triangles
 Three Similar Triangles (but differently oriented)
 Three Similar Triangles (but differently oriented) II
 Three Squares and Two Ellipses
 Three Tangent Circles
 Three Tangents Theorem
 Three Tangents Theorem II
 Three Tangents, Three Chords in Ellipse
 Three Tangents, Three Secants
 Toothpick Construction of a Square
 Touching Circles
 Touching Circles and Concurrency
 Touching Circles with Given Cnters II
 Transversal Trigonometry
 Triangle and Segment Count in a Polygon
 Triangle by HM segments
 Triangle Construction from Angle, Altitude and Median
 Triangle Geometry
 Triangle Sliding on Two Straight Lines
 Triangles, Squares and Areas from Temple Geometry
 Triangles with Equal Area
 Triangular Billiard
 Trigonometric Form of Ceva's Theorem (a tool for discovery of trigonometric identities and other problems)
 Tromino as a Reptile
 Tucker Circles
 Tucker Circles Through Homothety
 Twin Segments in Arbelos
 Two Butterflies Theorem
 Two Butterflies Theorem II
 Two Butterflies Theorem III
 Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
 Two Circles and One More
 Two Circles and Two More
 Two Circles in a Parallelogram
 Two Circles Inscribed in a Parallelogram
 Two Circles in a Square
 Two Circles in a Square II
 Two Circles in an Angle
 Two Circles on Angle Bisector
 Two Circles on a Side of a Triangle
 Two Circles, Tangents and Two Collinearities
 Two Circumcircles And Two Pairs of Parallels
 Two Circumcircles in Triangle
 Two Congruent Circles by Reflection
 Two Conics, Pascal, Chasles, and CrossRatio
 Two Homotheties in a Parallelogram
 Two Lines  Two Circles
 Two Pairs of Parallel Lines in a Triangle
 Two Pencils of Parallel Lines
 Two Proofs of MenelausTheorem
 Two Quadrilaterals
 Two Quadruplets of Concyclic Points
 Two Quadruplets of Tangent Circles
 Two Rectangles in a Rectangle
 Two Similar Triangles with Parallel Sides
 Two Simsons in a Triangle
 Two Squares and Another Square
 Two Triples of Similar Triangles
 Two Tangents to Parabola
 Two Triangles Inscribed in a Conic
 Two Triangles Inscribed in a Conic  with Elementary Solution
 Two Triangles Inscribed in a Conic  with Solution
 Two Triangles With Common Base and Altitude
 Two Triples of Concurrent Cevians and a Conic
 Two Triples of Concurrent Circles
 Van Schooten's Locus Problem
 Varignon Parallelogram
 VectenNikolin Hexagon
 Vecten's Collinearity
 Vecten's Mesh
 Vecten's Theorem
 Versatile Theorem
 Vietnamese Lemma towards an Extension of a Japanese Theorem
 Weitzenböck Inequality
 What does one see inside a spherical mirror?
 When a Triangle is Equilateral
 William Wallace's 1803 Statement of the Butterfly Theorem
 Wittenbauer's Parallelogram
 Y. Sawayama's Lemma
 Yet Another Concurrence on the 9Point Circle
 Yet Another Seven Circles Theorem
 Zaslavsky's theorem
Miscellaneous Demonstrations
 Calculation of the Digits of pi by the Spigot Algorithm
 Calculation of the Digits of pi (Faster Version)
 Convex Polygon Is the Intersection of Half Planes
 Disapperances à la Sam Loyd
 Disappearing Lines
 Disappearing Rectangles
 Distance Between Strings
 Experimentation with Dynamic Geometry Software: An Example
 Genetic Algorithm Solves the Toads and Frogs Puzzle
 Impossible Fork
 Impossible Frame
 Morphing
 Order and Chaos in Multiple Pendulums
 Projectile Trajectory
 Random Dot Stereograms
 Ratchet Effect
 Rotating Gears
 Sorting Algorithms
 Structural Constellation
 Watt's and Chebyshev's Linkage
