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Memory and Matching
Match answers
Memory - Easy
Memory - Medium
Memory - Hard
Arithmetic
1089 and a Property of 3-digit Numbers
Four Digits Magic Prediction
Five Digits Magic Prediction
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
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
Fixed Point of a Family of Parabolas
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
Square in a Circular Sector
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
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
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
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
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
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
5x5 Square Grid and 5 Circes
60
o
Angle And Importance of Being The Other End of a Diameter
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 Incenter Relative to Circumcircle
A Property of Isogonal Lines
A Property of Orthic Triangle
A Property of Perspective Triangles
A Straightedge Construction of the Midpoint of a Chord Common to Two Circles
A Theorem of M. Mansion
A Three Pegs Question
A Triangle With a 45 Degrees Angle in Square
Acute Angle Bisectors in a Right Triangle
Al-Nayrizi's Construction
An Appearance of a 30-60-90 Triangle
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 Nine-Point 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 And Perpendicular Segments in a Square
Equal Areas in Regular 2n-gons
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
Eye-to-Eye Theorem I
Eye-to-Eye 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
Mickey Might Be a Red Herring in the Mickey Mouse Theorem
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
One More Property of Equilateral Triangles
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
Playing on Flanks
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
Semi-regular Tessellation on Hinges A
Semi-regular Tessellation on Hinges B
Semi-regular 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, 45 Degrees Angle and Pythagoras' Look-alike
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
The Shortest Distance in a Circular Segment
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 Khea's Quickie
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
Tetrahedron
Octahedron
Cube
Icosahedron
Dodecahedron
Cube In Octahedron
Octahedron In Cube
Octahedron In Tetrahedron
Dissection of a Square Pyramid
Icosahedron In Cube
Tetrahedron In Cube
Other Solids
Császár Polyhedron 1
Császár Polyhedron 4
Great Stellated Dodecahedron
Mikhalev's Octahedrons
Pentagonal Prism
Square Pyramid
Square Toroid
Triangular Prism
Twisted Triangular Prism
Lennes' Polyhedron
Szilassi Polyhedron
Triangulated Dinosaur
Possible or Impossible?
Serendipitous Beauty
Stereographic Projection
Three Pyramids are Better Than Two
Outline (Fill-It-In) Mathematics
A bunch of fill-it-in outline problems and solutions
Interactive Word Problems
Interactive Logic Problems
Interactive Arithmetic Problems
Number Theory Problems
Fill-it-in Geometry Problems
Arithmetic Practice Problems
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
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
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 Pentagon with Right Angles
A Problem in Pentagon with Supplementary Angles
A Problem in Three Pentagons
A Property of Angle Bisectors
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 60° Rhombus in a 60° Isosceles Trapezoid
A Problem in a Configuration of Three Squares
A Stronger Triangle Inequality
A Triangle in a Rhombus with a 60 Degrees Angle
A Triangle of Antreas Hatzipolakis
Abutting Equilateral Triangles on a Line I
Adams' Circle
Addition of Radius-Vectors
Altitudes and the Power of a Point
An Equilateral Triangle In Parallel Lines
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 9-Point 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 9-Point 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
Asymmetric Propeller of Squares
Asymmetric Propeller of Squares II
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
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 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
Crossed-Lines 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
)
Droz-Farny Circles
Droz-Farny Line Theorem
Droz-Farny 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 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 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 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
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 9-Point 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 3-Bar Motion
Further properties of Van Aubel Configuration
Fuss' Theorem
Garcia's Archimedean Quadruplets
Garcia-Feuerbach Collinearity
Generalizing a Romanian Olympiad Problem
Geometric/Trigonometric Observation/Exercise
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 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 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
Luca Moroni's Problem In Equilateral Triangle
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 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 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
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
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 between Two Roads
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 6-Parpolygon
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
See-Saw 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 9-Point 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 Collinearity
Spiral Similarity and Collinearity
Square from Four Points, One on Each Side
Solution 1
Solution 2
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
Stan Fulger's Observation in Right Triangle
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
Thanos Kalogerakis' Collinearity in Triangle
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 on a Rectangular Hyperbola
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 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 Cross-Ratio
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
Vecten-Nikolin 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 9-Point 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
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Copyright © 1996-2018
Alexander Bogomolny
