Memory and Matching
- Match answers
- Memory - Easy
- Memory - Medium
- Memory - Hard
Arithmetic
- 1089 and a Property of 3-digit Numbers
- 2 Pails Puzzle
- 3 Jugs Puzzle
- 3 Jugs Puzzle in Barycentric Coordinates
- 3-Term 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
- EAN-13 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
- Ambassadors at a Round Table
- Averages, Arithmetic and Harmonic Means
- Base (Binary, Decimal, etc.) Converter
- Binary Color Device
- Breaking Chocolate Bars
- Candy Game: Integer Iterations on a Circle
- 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
- 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
- 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
- Self-descriptive Strings
- Self-documenting 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
- 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
- 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
- Müller-Lyer 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, V-Variant
- 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
- 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
- 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
- Schwarz Lantern
- Taylor Series Approximation to Cosine
- 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
- 3-Colors 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
- Life-like Automaton With Definable Rules
- Light Bulbs in a Circle
- Listing All Combinations
- Listing All Permutations
- Listing All Permutations: Johnson-Trotter 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 L-Trominoes
- 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
- Take-Away Games
- Turning Turtles
- Wythoff's Nim
- Wythoff's Nim II
Puzzles & Games
- 3-Chain
- 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 L-Trominoes
- 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
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
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
- Huntington-Hill Apportionment Method
- Jefferson's Apportionment Method
- Method of Markers II
- Shapley-Shubik Power Index Calculator
- Social Choice
- Webster's Apportionment Method
- Weighted Voting and Power Indices
Elementary Geometry
- A Characterization of the Euler Line
- A Hinged Realization of a Plane Tessellation
- A Lemma of Equal Areas
- A Multiplicative Identity of Areas in a Triangle
- A Property of Equiangular Polygons
- A Three Pegs Question
- Acute Angle Bisectors in a Right Triangle
- Al-Nayrizi's Construction
- 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
- Antiparallel
- Antiparallel via Three Reflections
- Archimedes' Law of the Lever
- Area of Median Triangle
- Area of Parallelogram
- Area of the Union of Two Squares
- Area of Triangle
- Equal Areas in Regular 2n-gons
- Areas in Square by Dissection
- Arithmetic in Disguise
- Between Major and Minor Circles
- Building a Bridge
- Building Bridges
- Carpet With a Hole
- Circle and Hyperbola as Lighthouse Curves II
- Circle Chains on Napoleon Triangles
- Circles and Semicircles in Rectangle
- Circle in a Square Inscribed in a Circle
- Circles with Equal Collinear Chords
- Circumcenter on Angle Bisector
- Classification of Quadrilaterals
- Common Chord and a Tangent
- Concyclicity in Rectangle
- Conic Sections as Loci of Points
- Construction of the Angle Bisector
- Construction of the Perpendicular Bisector
- Cross Points in a Polygon
- 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 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 Parallel Lines
- 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
- Fibonacci Bamboozlement
- Famous Theorem
- Finding Circle Center By Ruler Alone
- Fixed Point in Isosceles and Equilateral Triangles
- Focal Definition of Ellipse
- Focus and Directrix of Ellipse
- Four Hinged Squares
- Geoboard
- Half Turn, Reflection in Point
- Harmonic Mean Sangaku
- Hinged Greek Cross Tessellation
- Hippocrates' Squaring of a Lune
- Hunting Right Angles
- Huygens' proof of the Pythagorean Theorem
- Inscribed and Central Angles in a Circle
- Inscribed Angles
- Inscriptible Quadrilateral: An Illustration
- Intersections of a Circle with the Four Quadrants
- Invariance in Orthodiagonal Quadrilaterals
- Join Circles by Given Segment
- Lean Napoleon's Triangles
- Maimed Cake
- Mathematics in Pizzeria
- Napoleon's Theorem: A third proof with complex numbers
- Napoleon's Theorem via Two Rotations
- Napoleon on Hinges
- Nine Point Circle: an Elementary Proof
- Octagon in Parallelogram
- Octagon in Parallelogram: Overlapping Regions
- Olof Hanner's Jigsaw Puzzle
- Orthocenters of Two Triangles Sharing Circumcenter and Base
- Pappus' Generalization of Euclid I.47
- Orthodiagonal and Cyclic Quadrilaterals
- Parallel Chords II
- Parallel Chords in Crossing Circles
- Parallel Lines in a Quadrilateral
- Pascal: Necessary and Sufficient
- Polygon Metamorphosis
- Pompeiu's Theorem
- Property of Angle Bisectors
- Property of Rhombi
- Property of Semicircles
- 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
- Rhombus in a Cyclic Quadrilateral
- Right Triangles on Sides of a Square
- Sangaku with Angle between a Tangent and a Chord
- Secant Angles
- Secant Angles II
- Secant, Normal, Tangent
- Segment Trisection Induced by Parallels to Medians
- Semi-regular Tessellation on Hinges A
- Semi-regular Tessellation on Hinges B
- Semi-regular Tessellation on Hinges C
- Shearing a Polygon into a Triangle of Equal Area
- Simple Quadrilaterals Tessellate the Plane
- Shearing and Translation in Pythagorean Pants
- Square From Nowhere
- Square in a Circle Inscribed in a Square
- Square in Chair
- Squaring a Rectangle
- Three Touching Circles
- Translated Triangles
- Translation Transform
- Triangle Areas in a Parallelogram
- Triangle Areas in a Parallelogram II
- Triangle Classification
- Trigonometric Functions
- Two Altitudes, One Midpoint
- Two Color Coloring of the Plane
- Two Equilateral Triangles
- Two Intersecting Circles
- Two Parallels in a Triangle and One More
- 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
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
Outline (Fill-It-In) Mathematics
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A bunch of fill-it-in outline problems and solutions
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Geometry
- 120° Breeds 90°
- 2N-Wing Butterfly Theorem
- 3 Isosceles Trapezoids
- 5-Star and A Circle
- 6 to 9 Point Circle
- 60° Breeds 90°
- 9-point Circle as a Locus of Concurrency
- 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 Line in Triangle Through the Circumcenter
- A Parallelogram in Triangle
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku: Two Unrelated Circles
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Stronger Triangle Inequality
- A Triangle of Antreas Hatzipolakis
- Adams' Circle
- Addition of Radius-Vectors
- Altitudes and the Power of a Point
- An Isoperimetric Theorem
- 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 Pair of Twins in an Arbelos
- Anticomplementary Triangle Surprise
- Apollonian Circles Theorem
- Apollonius Problem
- Application of Ceva's Theorem
- 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 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
- Bevan's Point and Theorem
- Bisectal Circle
- 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
- Butterfly in Ellipse
- 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
- 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 Nest
- Cevian Triangle
- Chain of Circles on a Chord
- Chain of Inscribed 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 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
- Circle-Stacking 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 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
- Collinearity via Concyclicity
- Collinearity with the Orthocenter
- Common Tangents to Two Circles I
- Common Tangents to Two Circles II
- Complete Quadrilateral
- Concyclic Points in Equilateral Bumps
- Concurrency in Analytic Geometry
- Concurrency in Arbelos
- Concurrency in the Intouch Triangle
- Concyclic Circumcenters: A Dynamic View
- Concyclic Points in a Triangle
- Concyclic Points in Bride's Chair
- Conjugate Diameters in Ellipse
- Construction of Conics from Pascal's Theorem
- Consecutive Isosceles Decomposition
- Conway's Circle
- Copernicus' Theorem
- Crossed-Lines Construction of Shapes of Constant Width
- Cyclic Hexagon
- Cyclic Incenters
- Cyclic Quadrilateral, Concurrent Circles and Collinear Points
- Cycloids
- David Richeson's Extention of an Old Japanses Theorem
- Diameters and Chords
- Directly Similar Figures
- Desargues' Theorem
- Discovery of Duane DeTemple
- Dissection of Triangle into Rhombus
- Dissection of Triangle into Rhombus by Hubert Shutrick
- 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)
- Droz-Farny Circles
- Droz-Farny Line Theorem
- Dual to Pappus' Theorem
- Dynamic Construction of Ellipse and Other Curves
- Easy Construction of Bicentric Quadrilateral
- Easy Construction of Bicentric Quadrilateral II
- Eight Point Circle
- Ellipse, how to draw
- Ellipse in Arbelos
- Ellipse Between Two Circles
- 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 3-4-5 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 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
- Extouch Triangle in Poncelet Porism
- Eyeball Theorem
- 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 point and 9-point 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
- Find a Common Chord of Given Length
- 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
- Foot of Altitude and Minimum Distance
- Four 9-Point Circles in a Quadrilateral
- Four Centroids and Parallels
- Four Circles In a Triangle
- Four Concurrent Lines in a Cyclic Quadrilateral
- Four Incident Circles
- Four Crossing Circles
- Four Incircles in Equilateral Triangle
- Four Pedal Circles
- Four Touching Circles
- Four Touching Circles II
- Four Touching Circles III
- Four Triangles, One Circle
- Four Turtles (or Bugs, or Dogs, etc.)
- Friendly Kiepert's Perspectors
- From Foci to a Tangent in Ellipse
- Fundamental Theorem of 3-Bar Motion
- Fuss' Theorem
- 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
- Hexagon Parallel to Medial Triangle
- Hexagon Parallel to Orthic Triangle
- Hinged Squares
- Hjelmslev Theorem
- Hofstadter Triangles and Points
- Homologous Lines under Three Spiral Similarities
- Homothety
- Homothety between In- and Excircles
- 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
- 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
- Isogonal and Isotomic Conjugalities
- Isogonal Concurrency
- Isogonal image of the circumcircle
- Isosceles on the Sides of a Triangle
- Isotomy and Isogonality Hand-in-Hand
- 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 Theorem
- Kürschak's Tile and Theorem
- Law of Cosines
- Law of Cosines: PWW
- Law of Cosines: An Unfolded Variant
- La Hire's Theorem
- La Hire's Theorem, a Variant
- La Hire's Theorem in Ellipse
- Lighthouse Curves
- Line, Circle, and Fixed Points
- 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
- Maxwell's Theorem
- Median in Touching Circles
- Medians in a Quadrilateral
- Menelaus and Ceva Theorems
- Menelaus From Ceva
- Midline in Similar Triangles
- Midpoints and Orthogonality in Isosceles Triangles
- Midpoints from Gergonne Triangle
- Midpoints of the Lines Joining In- and Excenters
- Minkowski's addition of convex shapes
- Miquel's Point
- Miquel's Point of a 4-line 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 Theorem
- Morley's Theorem, Cardioids and Such
- Morley's Theorem, G. Zsolt Kiss' proof
- Nagel Line
- Nagel point
- Nagel Point of Medial Triangle
- Nagel's Theorem
- Napoleon-Barlotti 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
- Non-classic Way To Draw Cardioid
- No-Pedal Collinearity
- On Bottema's Shoulders
- On Bottema's Shoulders II
- One Sided Surface in 4D
- Optical Property of Ellipse
- Optimization Problem in Acute Angle
- Orthocenter and Three Equal Circles
- Orthocentric System From Rectangle
- 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
- 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
- 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 Polygons
- Pedal Triangle and Isogonal Conjugacy
- Pedoe's Theorem
- Pentagon in a Semicircle
- Perigal's Proof of the Pythagorean Proposition
- 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 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
- 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 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 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 6-Parpolygon
- Property of Angle Bisectors II
- Property of Circumscribed Quadrilaterals
- Property of Points Where In- and Excircles Touch a Triangle
- Property of the Line IO
- Property of the Line IO II
- Property of the Line IO: a proof from the Book
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
- MathPro Logo
- 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
- 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
- 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
- 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
- See-Saw Lemma
- Segment Trisection
- 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 9-Point Circles in Cyclic Quadrilateral
- Simultaneous Diameters in Concurrent Circles
- sin(N)
- Six Circles Theorem (Bui Quang Tuan)
- Six Circles Theorem (Elkies)
- Six Circles Theorem (Evelyn)
- Six Concyclic Points
- Six Concyclic Points II
- Six Incircles in an Equilateral Triangle
- Six Point Circle
- SixPoints, ThreeLines
- Soddy Circles and David Eppstein's Centers
- Some Properties of Napoleon's Configuration
- Sperner's Lemma
- Spira Mirabilis - Wonderful Spiral
- Spiral Similarity
- 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 Quadrilateral
- Squinting Eyes Theorem
- Star Construction of Shapes of Constant Width
- Steiner's Chain
- Steiner's porism
- Straight Edge Construction of Polar
- Symmedian and Antiparallel
- Symmedian and 2 Antiparallels
- Symmedian and the Tangents
- Symmedian in a Right Triangle
- 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 Plain Butterfly Theorem
- 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
- 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 Parabola Tangents
- 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
- Triangle and Segment Count in a Polygon
- 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 Rep-tile
- Tucker Circles
- Tucker Circles Through Homothety
- Twin Segments in Arbelos
- Two Butterflies Theorem
- Two Butterflies Theorem II
- Two Butterflies Theorem III
- 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 Circumcircles And Two Pairs of Parallels
- Two Circumcircles in Triangle
- Two Congruent Circles by Reflection
- Two Homotheties in a Parallelogram
- Two Lines - Two Circles
- Two Proofs of MenelausTheorem
- Two Quadrilaterals
- Two Quadruplets of Tangent Circles
- Two Rectangles in a Rectangle
- Two Simsons in a Triangle
- Two Squares and Another Square
- Two Triples of Similar Triangles
- Two Tangents to Parabola
- Two Triangles With Common Base and Altitude
- Two Triples of Concurrent Circles
- Van Schooten's Locus Problem
- Varignon Parallelogram
- Vecten's Collinearity
- Vecten's Mesh
- Vecten's Theorem
- Versatile Theorem
- Weitzenböck Inequality
- What does one see inside a spherical mirror?
- When a Triangle is Equilateral
- Wittenbauer's Parallelogram
- Y. Sawayama's Lemma
- Zaslavsky'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
- One Sheet Hyperboloid
- Platonic Solids - Regular Polyhedra
- Other Solids
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