Geometry articles, theorems, problems
Note that in the list below the pages that carry Java applets are designated with (*).
- 1 + 27 = 12 + 16 Sangaku
- 120° Breeds 90° (*)
- 3 Roads, 3 Travelers (*)
- 3 Utilities Puzzle
- 3D Quadrilateral - a Coffin Problem
- 3-4-5 Triangle by a Kid
- 4 Travelers problem
- 6 to 9 Point Circle (*)
- 60o Breeds 90o (*)
- 7 = 2 + 5 Sangaku
- 9-point Circle as a Locus of Concurrency (*)
- A better solution to a difficult sangaku problem
- A Case of Similarity (*)
- A Circle in an Isosceles triangle from the 2006 Itish MO
- A Circle Related to Incenter and Circumcenter (*)
- A Circle Rolling in an Equilateral Triangle (*)
- A Circle With Two Centers (*)
- A Geometric Limit
- A Hard but Important Sangaku
- A Lemma of Equal Areas (*)
- A Line in a Square Grid (*)
- A Line in Triangle Through the Circumcenter (*)
- A Parallelogram in Triangle (*)
- A Possibly First Proof of the Concurrence of Altitudes
- A Problem In an Isosceles Right Triangle (*)
- A Problem on an Icy Cone
- A problem with equilateral triangles (*)
- Properties of the Figures in Euclid I.47
- A Property of Equiangular Polygons (*)
- A Property of Rhombi (*)
- A Property of the Line IO (*)
- A Property of the Line IO: A Bundled Statement and Proof (*)
- Property of the Line IO: a proof from the Book (*)
- A Relation in Triangle
- A Restored Sangaku
- 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 Sushi Morsel
- A Triangle of Antreas Hatzipolakis (*)
- A Trigonometric Identity in a Right Triangle from the 1960 IMO
- A Trigonometric Solution to a Difficult Sangaku Problem
- About a Line and a Triangle (*)
- Abul Wafa al-Buzjani's Proof of the Pythagorean Theorem (*)
- Adams' Circle (*)
- Addition and Subtraction Formulas for Sine and Cosine
- Addition and Subtraction Formulas for Sine and Cosine II
- All Triangles Are Isosceles (*)
- Altitudes (*)
- Altitudes and the Euler Line (*)
- Altitudes and the Power of a Point (*)
- An Extension of van Schooten's Theorem
- An Identity in a Regular Heptagon
- An Isoperimetric Theorem (*)
- An Old Japanese Theorem
- Angle: An Illustrated Classification (*)
- Angle Bisector in Equilateral Trapezoid
- Angle Bisector in Parallelogram (*)
- Angle Bisector in Touching Circles (*)
- Angle Bisectors (*)
- Angle Bisectors and Perpendiculars in a Quadrilateral (*)
- Angle Bisectors in Ellipse (*)
- Angle Bisectors in Ellipse II (*)
- Angle Bisectors in a Quadrilateral (*)
- Angle Bisectors On Circumcircle (*)
- Angle Subtended by a Diameter (*)
- Angle Preservation Property (*)
- Angle Trisection by Archimedes (*)
- Angle Trisection by Hippocrates (*)
- Angle Trisection by Paul Vjecsner (*)
- Angle Trisectors on Circumcircle (*)
- Angle Trisection by Paper Folding
- Angles in a Cube I (*)
- Angles in a Cube II (*)
- Angles in a Cube III (*)
- Angles in Triangle Add to 180o
- Anticomplementary Triangle Surprise (*)
- Antiparallel (*)
- APMO 1995 Problem 3
- APMO 1998, Problem 4
- APMO 2002, Problem 3
- APMO 2003, Problem 2
- APMO 2004, Problem 2
- Apollonian Circles Theorem (*)
- Apollonian Gasket (*)
- Apollonius Problem (*)
- Approximate Construction of Regular Pentagon by A. Dürer
- Arbelos -- the Shoemaker's Knife (*)
- Archimedes' Book of Lemmas
- Archimedes' Law of the Lever (*)
- Archimedes' Method
- Archimedes' Twins in the Edo Period
- Area of Median Triangle (*)
- Area of Parallelogram (*)
- Area of the Union of Two Squares (*)
- Area of Triangle (*)
- Areas and Centroid in a Triangle (*)
- Arithmetic-Geometric Mean Inequality
- Arithmetic Mean Sangaku (*)
- Arithmetic Mean Sangaku II (*)
- Around the Incircle (*)
- Around the Incircle in a Polygon (*)
- Artisans vs Mathematicians
- Asian Pacific Kite (*)
- Assimilation Illusion (*)
- Asymmetric Propeller (*)
- Barbier, The Theorem of (*)
- Barycentric coordinates
- Bender: A Visual Illusion (*)
- Between Major and Minor Circles (*)
- Bevan's Point and Theorem (*)
- Bimedians in a Regular Tetrahedron (*)
- Bisectal Circle (*)
- Bisecting arcs
- Bisecting a shape
- Bisector of an imaginary angle may be real
- Bottema Shatters Japan's Seclusion (*)
- Bottema's theorem (*)
- Bottles in a Wine Rack (*)
- Bounded Distance
- Brahmagupta's Theorem (*)
- Brianchon in Ellipse (*)
- Brianchon's theorem (*)
- Brick Wall Illusion (*)
- Bride's Chair (*)
- Brief introduction to Rational Trigonometry
- Broken Chord Theorem (*)
- Building a Bridge (*)
- Building Bridges (*)
- Bulging lines illusion (*)
- Butterfly Theorem
- Cabart's Collinearity
- Cantor Set and Function
- Cantor's Theorem (*)
- Carnot's Theorem
- Carnot's Theorem
- Carnot's Theorem (Generalization of Wallace's theorem) (*)
- Carpet With a Hole (*)
- Carpets Theorem (*)
- Centroid, a Characteristic Property Of
- Centroids in a Polygon (*)
- Chasing Angles in Pascal's Hexagon (*)
- Circle, Isosceles Triangle and Fixed Point (*)
- Circle in a Square Inscribed in a Circle (*)
- Circles on the sides of a triangle from the 2005 Austrian MO
- Circles Tangent to Circumcircle (*)
- Circles with Tangent Diameters (*)
- Ceva and Menelaus Meet on the Roads
- Ceva in Circumscribed Quadrilateral
- Ceva's Theorem
- Ceva's Theorem and Fibonacci Bamboozlement (*)
- Cevian Nest (*)
- Cevian Triangle (*)
- Chain of Inscribed Circles (*)
- Chaos, Emergence of (*)
- Chasles' Theorem
- Cherchez le quadrilatère cyclique (*)
- Cherchez le quadrilatère cyclique II (*)
- Chords, Concurrency and Orthic Triangle (*)
- Chvatal's Art Gallery Theorem (*)
- Circle and Hyperbola as Lighthouse Curves
- Circle and Hyperbola as Lighthouse Curves II
- Circle Crosses Triangle (*)
- Circle Concurrency and Spiral Similarity (*)
- Circle Division by Euclid and Leonardo (*)
- Circle of Similitude (*)
- Circle through the Incenter (*)
- Circle through the Incenter And Antiparallels (*)
- Circle-Stacking Theorem (*)
- Circles And Parallels (*)
- Circles and Semicircles in Rectangle (*)
- Circles in a Circular Segment
- Circles in Morley's Triangles (*)
- Circles Lined on the Legs of a Right Triangle (*)
- Circles On Cevians (*)
- Circles through the Orthocenter (*)
- Circles with Equal Collinear Chords (*)
- Circular Poggendorff Illusion (*)
- Circle Inscribed in a Circular Segment (*)
- Circumcenter on Angle Bisector (*)
- Circumcevian Triangle (*)
- Classification of Quadrilaterals (*)
- Cleavance Center (*)
- Cleaver (*)
- Clifford's Chain (*)
- Coaxal Circles on Perpendicular Bisector (*)
- Coaxal Circles Theorem (*)
- Collage Theorem (*)
- Collinearity in Bicentric Quadrilaterals (*)
- Color Cycling on the Mandelbrot Set (*)
- Common Chord and a Tangent (*)
- Common Tangents to Two Circles I (*)
- Common Tangents to Two Circles II (*)
- The Complete Quadrilateral (*)
- Complex Numbers and Geometry
- Concurrency in the Intouch Triangle (*)
- Concyclic Circumcenters: A Dynamic View (*)
- Concyclic Points in a Triangle
- Concyclic Points in Bride's Chair (*)
- Concyclic Points in Equilateral Bumps (*)
- Conic Sections
- Conic Sections as Loci of Points (*)
- Conjugate Diameters in Ellipse (*)
- Consecutive Isosceles Decomposition (*)
- Constant Chord
- Constructing a triangle from its angle bisectors is in general impossible
- Construction of a Square from the 2008 Austrian MO
- Construction of a Triangle from the Circumcenter, Orthocenter and Incenter
- Construction of a Triangle from Two Vertices and the Centroid
- Construction of Conics from Pascal's Theorem (*)
- Construction of Regular Pentagon by H. W. Richmond
- Construction of the Angle Bisector (*)
- Construction of the Perpendicular Bisector (*)
- Construction problem (*)
- Constructions Related To An Inaccessible Point
- Construction with the Compass Only
- Continuous Line Illusion (*)
- Convex Polygon Is the Intersection of Half Planes (*)
- Copernicus' Theorem (*)
- Construction with Optimization in Angle - Problem the 1979 USA Mathematical Olympiad
- cos(36°) = (1 + √5)/4
- Cosine Law
- Counting Triangles and Segments in a Polygon (*)
- Cross-Ratio
- Crossed-Lines Construction of Shapes of Constant Width (*)
- Crossing Number of a Graph
- Cube Counting Ambiguity (*)
- Curry's Paradox (*)
- Cut the Cone (*)
- Cut the Cube (*)
- Cut the Cylinder (*)
- Cutting Squares
- Cutting Triangles off Regular Hexagon
- Cyclic Hexagon (*)
- Cycloids (*)
- Day's Sine Illusion (*)
- Delboeuf Illusion (*)
- Delian Problem Solved (*)
- Desargues in the Bride's Chair (with Pythagoras)
- Desargues' Theorem (*)
- Determination of p
- Diagonal Count (*)
- Diagonals in a Cyclic Quadrilateral
- Diameters and Chords (*)
- Directly Similar Figures (*)
- Disappearing Lines puzzle (*)
- Discovery of Duane DeTemple (*)
- Dissection of a 10×13 Rectangle into Two Chessboards
- Dissection of Cyclic Quadrilateral (*)
- Distance
- Distance between the Orthocenter and Circumcenter
- Distance Formula
- Distorted Lines Illusion (*)
- Divergent Spirals Illusion (*)
- Divide a Circle into N Parts of Equal Area
- Divide a segment into n equal parts
- Dividing Evenly a Quadrilateral (*)
- Dividing Evenly a Quadrilateral II (*)
- Double Generation Theorem (*)
- Drawing circles (*)
- Droz-Farny Circles (*)
- Droz-Farny Line Theorem (*)
- Duplication of the Cube
- Dynamic Construction of Ellipse and Other Curves (*)
- E. W. Dijkstra's Proof of the Pythagorean Theorem (*)
- Easy Construction of Bicentric Quadrilateral (*)
- Easy Construction of Bicentric Quadrilateral II (*)
- Egyptian Triangle By Paper Folding
- Eight Equal Tangents (*)
- Eight Point Circle (*)
- Ellipse in Arbelos (*)
- Ellipse Between Two Circles (*)
- Equal Angles in Two Circles (*)
- Equal Angles in Two Circles II (*)
- Equal Areas in Equilateral Triangle (*)
- Equal Areas In Parallelogram à la Pythagoras (*)
- Equal Chords in Crossing Circles (*)
- Equal Circles, Medial Triangle and Orthocenter (*)
- Equal Incircles Theorem (*)
- Equichordal points: Just Do It (*)
- 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 and 3-4-5 Triangles (*)
- Equilateral Triangle on Parallel Lines (*)
- Equilateral Triangle, Straight Line and Tangent Circles (*)
- Equilateral Triangles and Incircles in a Square
- Equilateral Triangles on Diagonals of Antiequilic Quadrilateral (*)
- Equilateral Triangles On Sides of a Parallelogram (*)
- Equilateral Triangles On Sides of a Parallelogram II (*)
- Equilateral Triangles on Sides of a Quadrilateral (*)
- Equilic Quadrilateral I, a Flawed Proof (*)
- Equilic Quadrilateral I, A Variation (*)
- Equilic Quadrilateral II (*)
- The Erdös-Mordell Inequality
- Escher's Theorem (*)
- Euclid's Fifth Postulate
- Euclid's Elements Reference Page
- Euclidean Construction of Center of Ellipse (*)
- Euclidean Construction of Tangent to Ellipse
- Euler Line Cuts Off Equilateral Triangle (*)
- Euler Line and 9-Point Circle (*)
- Euler's Formula and Poncelet Porism
- Existence of the Circumcenter (*)
- Existence of the Euler Line: An Elementary Proof (*)
- Experimentation with Dynamic Geometry Software: An Example (*)
- Euclid I.43 (*)
- Euclid I.43 Extended (*)
- Exterior Angle Theorem - an appreciation
- External Angle Bisector (*)
- Extouch Triangle in Poncelet Porism (*)
- Eyeball Theorem (*)
- Fagnano's problem (*)
- Faulty Symmetry
- Fermat point and 9-point Centers (*)
- Feuerbach's Theorem: A Proof (*)
- Find a Common Chord of Given Length (*)
- Find the Angle! (*)
- Finding Circle Center By Ruler Alone (*)
- Five Concyclic Points (*)
- Five Incircles in a Square (*)
- Five Incircles Theorem (*)
- Fixed Point in a Kite - Problem from the 1993 Asian Pacific MO
- Fixed Point of Circles Orthogonal to the Given One (*)
- Fixed points from 2007 Irish Mathematical Olympiad
- Fleury's Algorithm and Euler's Paths and Cycles (*)
- Foot of Altitude and Minimum Distance (*)
- Focal Definition of Ellipse (*)
- Focus and Directrix of Ellipse (*)
- For every object there is a distance at which it looks its best.
- Ford's touching circles
- Four Centroids and Parallels (*)
- Four Circles In a Triangle (*)
- Four Concurrent Lines in a Cyclic Quadrilateral (*)
- Four Construction Problems (*)
- Four Crossing Circles (*)
- Four Geometric Problems of Antiquity
- Four Hinged Squares (*)
- Four Incident Circles (*)
- Four Incircles in Equilateral Triangle (*)
- Four 9-Point Circles in a Quadrilateral (*)
- 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.) (*)
- Fractal curves and dimension (*)
- 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 Illustration of a Convergent Series
- Geometric Mean Sangaku
- Geometric Meaning of the Geometric Mean
- Geometric Optimization from the Asian Pacific Mathematical Olympiad (*)
- Gergonne and Medial Triangles Are Orthologic (*)
- Gergonne and Soddy Lines Are Perpendicular (*)
- Gergonne in Ellipse (*)
- Gergonne and Van Obel Theorems
- Gion Shrine Problem
- Given Parabola, Find Axis
- Glide Reflection (*)
- Golden Ratio in Geometry
- Golden Ratio in an Irregular Pentagon
- Golomb's inductive proof of a tromino theorem (*)
- Gothic Arc (*)
- Graph Theory (*)
- Grebe: from Ladies' Diary to Carroll's Pillow (*)
- Grebe's Theorem
- Greg Markowsky's Problem for Parabola (*)
- Griffiths Points in Cyclic Quadrilateral (*)
- Half Turn, Reflection in Point (*)
- Halving a square
- Harmonic Mean Sangaku (*)
- Harmonic Ratio
- Harmonic Ratio in Complex Domain
- Hart Circle (*)
- Hart's Inversor (*)
- Haruki's Lemma (*)
- Haruki's Theorem
- Hausdorff Distance
- Hearty Munching on Cardioids (*)
- Height/Width Interplay Illusion (*)
- Hering Illusion (*)
- Hermann Grid Illusion (*)
- Heron's formula
- Heron's Formula: a Proof (*)
- Heron's Problem (*)
- Hexagon Parallel to Medial Triangle (*)
- Hexagon Parallel to Orthic Triangle (*)
- Hinged Squares (*)
- Hippocrates' Squaring of a Lune (*)
- Hippocrates' Squaring of Lunes
- Hjelmslev Theorem (*)
- Homologous Lines under Three Spiral Similarities (*)
- Homothety (*)
- Hooper's Paradox (*)
- Hunting Right Angles (*)
- Ian McGee's Observation (*)
- IMO 1998 Problem 1
- IMO 2004, Problem 5
- IMO 2006, Problem 1
- Impossible Fork (*)
- Impossible Frame (*)
- Importance of Occluders: Motion Binding (*)
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral (*)
- Incidence in Feuerbach's Theorem (*)
- Infinitude of Primes - A Topological Proof
- Inscribed and Central Angles in a Circle (*)
- Inscribed Angles (*)
- Inscriptible and Exscriptible Quadrilaterals (*)
- Inscriptible Quadrilateral: An Illustration (*)
- Internal Tangents to Three Circles (*)
- Intersecting Chords Theorem
- Intouch Triangle in Poncelet Porism (*)
- Invariant Area Sweep Implies Pythagorean Theorem (*)
- Inversion: Reflection in a Circle (*)
- Is Every Trapezoid Parallelogram? (*)
- Isosceles Tetrahedron
- Isogonal and Isotomic Conjugalities (*)
- Isogonal Concurrency (*)
- Isogonal image of the circumcircle (*)
- Isoperimetric Theorem and Inequality
- Isosceles on the Sides of a Triangle (*)
- Isotomy and Isogonality Hand-in-Hand (*)
- Iterations and the Mandelbrot Set (*)
- Iterations on a Circle Through Three Points (*)
- Iterations on Euler Lines (*)
- J. Casey's Additions to Euclid I.47
- Jan Stevens' proof of E. W. Dijkstra's Generalization of the Pythagorean theorem
- Joachimsthal's Notations
- John Sharp's Paradox (*)
- Join Circles by Given Segment (*)
- Joseph Keech in Bride's Chair (*)
- Judd Illusion (*)
- Julia sets indexing (*)
- Kanizsa Triangle (*)
- Kiepert's Theorem (*)
- Knots [avi movie]
- Kürschak's Tile and Theorem (*)
- La Hire's Theorem (*)
- Langman's Paradox (*)
- Lean Napoleon's Triangles (*)
- Length of the diagonal of the unit square equals the square root of 2
- Lepidoptera of the Circles (*)
- Line IO in Bicentric Quadrilaterals
- Line through a Point in an Angle
- Line, Circle, and Fixed Points (*)
- Locus of Points in a Given Ratio to Two Points (*)
- Longest segment
- MacLaurin's Construction of Conics (*)
- Maimed Cake (*)
- Malfatti's Problem (*)
- Malfatti's Problem, Hart's Solution (*)
- Mathematical Droodles (*)
- Mathematics in Pizzeria (*)
- MathPro Logo (*)
- Maximal Properties of the Pythagorean Relation
- Maximum Area Property of Equilateral Triangles
- Maximum Area Property of the Medial Triangle
- Maximum Perimeter Property of the Incircle {*}
- Maximum Ratio of Similarity - Belarus Olympiad Problem
- Maxwell's Theorem (*)
- May the Product of Planes Be a Sphere?
- Mechanical Proofs of the Pythagorean Theorem
- Median in Touching Circles (*)
- Medians
- Medians in a Quadrilateral (*)
- Menelaus Theorem (*)
- Metric Relations in a Triangle
- Midline in Similar Triangles (*)
- Midpoints from Gergonne Triangle (*)
- Midpoints of the Lines Joining In- and Excenters (*)
- Minesweeper Theorem
- 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 (*)
- Moebius Strip [avi movie]
- Möbius' Strip (*)
- Models and Metamathematics
- Monge via Desargues (*)
- Monge via Desargues II (*)
- Morley Constellation (*)
- Morley's Miracle (*)
- Morphing (*)
- Müller-Lyer Illusion (*)
- Munching on Circles
- Munching on Inscribed Angles
- My logo
- Nagel Line (*)
- Nagel point (*)
- Nagel Point of the Medial Triangle (*)
- Nagel's Theorem (*)
- Napoleon's Theorem
- Necker Cube (*)
- Neuberg Sangaku (*)
- Newton's Construction of Conics (*)
- Newton's Theorem (*)
- Nine Point Circle (*)
- Nine Point Circle: an Elementary Proof (*)
- No Equilateral Triangles, Please (*)
- Nobbs' Points, Gergonne Line (*)
- Non-Eclidean Geometries
- No-Pedal Collinearity (*)
- Occluded Wheel Illusion (*)
- Olof Hanner's Jigsaw Puzzle (*)
- On the Difference of Areas
- Optimization Problem in Acute Angle (*)
- Octagon in Parallelogram: Overlapping Regions (*)
- Octagon in Parallelogram (*)
- Optical Property of Ellipse (*)
- Orbison illusion (*)
- Orthocenter and Three Equal Circles (*)
- Orthocenters
- Orthocenters of Two Triangles Sharing Circumcenter and Base (*)
- Orthocentric System From Rectangle (*)
- Orthodiagonal and Cyclic Quadrilaterals (*)
- Orthogonality in Isosceles Triangles (*)
- Orthologic Triangles in a Quadrilateral (*)
- Orthopole (*)
- Out of Pentagon Sangaku
- Packing shapes in a plane
- Pairs of Homologous Lines under Spiral Similarities (*)
- Pairs of Incircles in a Quadrilateral (*)
- Paper Folding & Cutting Sangaku
- Paper Folding Geometry
- Paper strip activities [avi movies]
- Paper strip activities, two strips
- Pappus' Generalization of Euclid I.47 (*)
- Pappus' Theorem (*)
- Parabola (*)
- Parabolic Mirror, Illustration (*)
- Parabolic Mirror, Theory (*)
- Paragon (*)
- Parahexagon (*)
- Parallel Chords (*)
- Parallel Chords II (*)
- Parallel Chords in Ellipse (*)
- Parallel Chords in Crossing Circles (*)
- Parallel Lines in a Quadrilateral (*)
- Parallelogram and Similar Triangles (*)
- Parallelogram Illusion (*)
- Parallelogram Iterations (*)
- Parallelogram Law (*)
- Parallelogram Law: A PWW (*)
- Parallelogram Law: A Tessellation (*)
- Partition of point sets in a plane
- Partitioning 3-Space with Circles
- Partitioning a Circle
- Pascal: Necessary and Sufficient (*)
- Pascal in a Cyclic Quadrilateral (*)
- Pascal Lines: Steiner and Kirkman Theorems (*)
- Pascal Lines: Steiner and Kirkman Theorems II (*)
- Pascal's theorem (*)
- Peacock's Tail Sangaku
- Pedal Collinearities (*)
- Pedal Polygons (*)
- Pedal Triangle and Isogonal Conjugacy (*)
- Pedoe's Theorem (*)
- Pencils of Cubics
- Pentagon, construction of
- Pentagon Proportions Sangaku
- Periodic patterns in sin(N) (*)
- Perpendicular Bisectors in an Inscriptible Quadrilateral (*)
- Perpendicular Bisectors in an Inscriptible Quadrilateral II (*)
- Perpendicular lines in a cyclic quadrilateral from 1975 Canadian Mathematical Olympiad
- Pivot Theorem (*)
- Plane Filling Curves (*)
- Plane Isometries
- Plato's Geometric Numbers: can you better Sam Loyd?
- Peaucellier Linkage (*)
- Pick's Theorem (*)
- Poggendorff's Illusion
- Point common to two similar rectangles (*)
- Point in a Square
- Points Generated by the Nine Points
- Polar Circle (*)
- Poles and Polars (*)
- Polygons: formality and intuition (*)
- Polygons Formed by Perpendicular Bisectors (*)
- Poncelet's Porism (*)
- Poncelet's Porism in Ellipses (*)
- Poncelet's Theorem (*)
- Ponzo Illusion (*)
- Ponzo Illusion II (*)
- Possible or Impossible? (*)
- Power of a point with respect to a circle
- 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 2009
- Problem in an Equilateral Triangle (*)
- Problem in an Equilateral Triangle II (*)
- Problem in Three Squares (*)
- Problem of Equal Steps (*)
- Problem of Equal Steps II (*)
- Problem on a Pentagonal Star
- Problem with angle bisector from 1975 Canadian Mathematical Olympiad
- Problem with equal angles from 1987 IberoAmerican Mathematical Olympiad
- Projections on Internal and External Angle Bisectors (*)
- Projective Collinearity in a Quadrilateral (*)
- Projective infinity
- Properties of the Circle of Similitude (*)
- Property of 6-Parpolygon (*)
- Property of Angle Bisectors (*)
- Property of Angle Bisectors II (*)
- Property of Circumscribed Quadrilaterals (*)
- Property of Points Where In- and Excircles Touch a Triangle (*)
- Ptolemy's Theorem
- Ptolemy on Hinges
- Puzzles on graphs.
- Pythagoras' Theorem (87 proofs)
- Pythagoras and Vecten Break Japan's Isolation (*)
- Pythagorean Triples (*)
- Quadrature: A Child's Play
- Quadrilateral With Equal Opposite Sides And Angles (*)
- Quadrilaterals Formed by Perpendicular Bisectors (*)
- 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 (*)
- Random Dot Stereograms (*)
- Ratios and Orthogonality in a Triangle - Problem from the 2006 Romania MO
- Rational Points On a Circle
- Reflection In Line (*)
- Reflections in Ellipse (*)
- Reflections of a Line Through the Orthocenter (*)
- Reflections of a Point on the Circumcircle (*)
- Reflections of the Circumcenter
- Reflections of the Orthocenter (*)
- Reflections of the Orthocenter II
- Regular Polyhedra - Platonic Solids
- Other Solids
- Regular polyhedra and Euler's Theorem
- Relations in a Cyclic Polygon
- Remarkable Line in a Quadrilateral (*)
- Remarkable Line in Cyclic Quadrilateral (*)
- Rendezvous (*)
- Reverse Spoke Illusion (*)
- Revolving Circles Illusion (*)
- Rhombus in a Cyclic Quadrilateral (*)
- Riemann Sphere and Möbius Transformation
- Rotating Top Illusion (*)
- Rotating Gears (*)
- Rotation Transform (*)
- Rouse Ball's Fallacy (*)
- Rusty Compass Construction of Equilateral Triangle (*)
- Salinon: From Archimedes' Book of Lemmas (*)
- Sam Loyd's Geometric Puzzle
- Sam Loyd's Son's Dissection (*)
- Sangaku: Reflections and the Phenomenon
- Sangaku: Critique of My View and a Response
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan? (*)
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord (*)
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles (*)
- Sangaku with Versines (*)
- Sangakus with a Mixtilinear Circle (*)
- Scintillating Grid Illusion (*)
- Secant Angles (*)
- Secant Angles II (*)
- Secant, Normal, Tangent (*)
- Secant, Tangents and Orthogonality (*)
- See-Saw Lemma (*)
- Segment Trisection (*)
- Separating point sets with a circle
- Sequences of Touching Circles
- Seven Circles Theorem (*)
- Shaggy Dog Theorem
- Shapes in a lattice
- Shapes of constant width (*)
- Shearing and Translation in Pythagorean Pants (*)
- Shearing a Polygon into a Triangle of Equal Area (*)
- Shearing Butterflies in Quadrilaterals (*)
- Shepard's Parallelogram Illusion (*)
- Shifting Cube Illusion (*)
- Shimmer Illusion (*)
- Short Construction of the Geometric Mean
- Shortest Fence in a Quarter-Circle Pasture
- Shredding the torus
- Sierpinski Gasket By Common Trema Removal (*)
- Sierpinski Gasket Via Chaos Game (*)
- Sierpinski Gasket by Trema Removal (*)
- Similar Triangles And More
- Similar Triangles on Sides of a Quadrilateral (*)
- Similar Triangles on Sides and Diagonals of a Quadrilateral (*)
- Simple dissection of an acute triangle
- Simple Quadrilaterals Tessellate the Plane (*)
- Simson Line (*)
- Simsons and 9-Point Circles in Cyclic Quadrilateral (*)
- Simultaneous Contrast Illusion (*)
- Simultaneous Contrast Illusion II (*)
- Simultaneous Contrast: Koffka's Illusion (*)
- Simultaneous Contrast: White's Illusion (*)
- Simultaneous Contrast: White's Illusion, V-Variant (*)
- Simultaneous Generalization of the Theorems of Ceva and Menelaus
- Sine and Cosine of 15 Degrees Angle
- Sine, Cosine, and Ptolemy's Theorem
- 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 (*)
- Spiral Similarity (*)
- Square and Circle in a Gothic Cupola
- Square From Nowhere (*)
- Square in a Circle Inscribed in a Square (*)
- Square in a Right Triangle (*)
- Square in Chair (*)
- Square Inscribed in a Triangle (*)
- Square Inscribed in a Triangle II (*)
- Squares in Circles (*)
- Squares Inscribed In a Triangle III (*)
- Squares on Sides of a Quadrilateral (*)
- Square out of a Quadrilateral
- Squaring a circle
- Squaring a Rectangle (*)
- Squinting Eyes Theorem (*)
- SSA
- SSS
- Staircase Illusion (*)
- Star of David (*)
- Steiner's Sangaku
- Steiner's Theorem
- Stereographic Projection (*)
- Stereographic Projection and Inversion
- Stereographic Projection and Radical Axes
- Stewart's theorem
- Straight Edge Only Construction of Polar
- Structural Constellation (*)
- Sum of Squares in Equilateral Triangle
- Sum of Squares of Distances to Vertices
- Symmedians
- Symmetries in a Triangle
- Tangency Point of Two Circles (*)
- Tangent and Secant (*)
- Tangent as a Radical Axis (*)
- Tangent Circles and an Isosceles Triangle (*)
- Tangent Circles in a Parallelogram (*)
- Tangent in Concurrency (*)
- Tangent Lines and Circles in Convex Quadrilateral (*)
- Tangent to two circles
- Tangents and Diagonals in Cyclic Quadrilateral (*)
- Tangents, Perpendiculars and Geometric Mean (*)
- Tarski-Banach Decompositions
- Taylor circle (*)
- Tesseract (*)
- The 80-80-20 Triangle
- The Chaos Game: Address Space vs IFS (*)
- The only rational triangle
- Thébault's Problem I (*)
- Thébault's Problem II (*)
- Thébault's Problem III (*)
- Thébault's Problem III, Proof (*)
- Thébault's Problem IV (*)
- Theorem of Complete Quadrilateral (*)
- Theorems of Ceva and Menelaus, an Illustrated Generalization (*)
- Three Circles and Common Tangents
- 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 Equal Circles in a Semicircle
- Three Incircles In a Right Triangle (*)
- Three Incircles in a Triangle (*)
- Three Isosceles Triangles (*)
- Three Orthogonal Circles Through Three Given Points (*)
- Three Parallels in a Triangle (*)
- Three Points on a Parabola (*)
- Three Pyramids are Better Than Two (*)
- Three regions sharing a common boundary
- Three Similar Polygons (*)
- Three Similar Triangles (*)
- Three Similar Triangles II (*)
- Three Squares and Two Ellipses (*)
- Three Tangent Circles (*)
- Three Tangents, Three Chords in Ellipse (*)
- Three Tangent Circles Sangaku
- Three Tangents Theorem (*)
- Three Tangents, Three Secants (*)
- Three Touching Circles (*)
- Toothpick Construction of a Square (*)
- Topological preliminaries
- Touching Circles (*)
- Touching Circles and Concurrency (*)
- Touching Circles with Given Centers II (*)
- Train wheel quandary
- Transitivity in Action (*)
- Translated Triangles (*)
- Translation Transform (*)
- Triangle ABC is right iff sin²A + sin²B + sin²C = 2
- Triangle Areas in a Parallelogram (*)
- Triangle Areas in a Parallelogram II (*)
- Triangle Classification (*)
- Triangle constructions and facts (*)
- Triangle Sliding on Two Straight Lines (*)
- Triangles, Squares and Areas from Temple Geometry (*)
- Triangles with Equal Area (*)
- Triangular Billiard (*)
- Triangulating Squares
- Trigonometric Form of Ceva's Theorem (*)
- Trigonometric Functions (*)
- Trigonometric Proof of the Pythagorean Theorem
- Trigonometry by Paper Folding
- Trisectable angle that is not constructible
- Tucker Circles (*)
- Tucker Circles Through Homothety (*)
- Twelve Matchsticks Area Puzzle
- Twisted Cord Illusion (*)
- Two Altitudes, One Midpoint
- Two Arbelos, Two Chains
- 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 coins puzzle
- Two Congruent Circles by Reflection (*)
- Two Color Coloring of the Plane (*)
- Two Equilateral Triangles (*)
- Two Homotheties in a Parallelogram (*)
- Two Intersecting Circles (*)
- Two Lines - Two Circles (*)
- Two Parallels in a Triangle and One More (*)
- Two Perpendiculars From a Point to a Line (*)
- Two Quadrilaterals (*)
- Two Quadruplets of Tangent Circles (*)
- Two Sangaku with Equal Incircles
- Two Simsons in a Triangle (*)
- Two Squares and Another Square (*)
- Two Symmetric Triangles Are Directly Decomposable (*)
- Two Touching Circles (*)
- Two Triangles With Common Base and Altitude (*)
- Two Triples of Concurrent Circles (*)
- Two Triples of Similar Triangles (*)
- Unfolding Spirals Illusion (*)
- Urquhart's Theorem
- USA 1975 Problem 4
- Use of Homogeneous Coordinates for Geometrical Calculations
- Van Obel Theorem, Determinants, and Barycentric Coordinates
- Van Schooten's and Pompeiu's Theorems (*)
- Van Schooten's Locus Problem (*)
- Variations on the Theme of Tremas (*)
- Variations on the Theme of Tremas II (*)
- Varignon Parallelogram (*)
- Various Geometries
- Vecten's Collinearity (*)
- Vecten's Mesh (*)
- Vector Addition and Subtraction [Flash]
- Vector product
- Versatile Theorem (*)
- Viewing a Statue
- Viviani's Theorem (*)
- Viviani's Theorem II (*)
- Wallace-Bolyai-Gerwien Theorem
- Wallace's Theorem (*)
- Weird curves bound normal areas
- Weitzenböck Inequality (*)
- What does one see inside a spherical mirror? (*)
- What Is Analytic Geometry?
- What Is Angle?
- What Is Ellipse?
- What Is Line?
- What Is Point?
- When a Triangle is Equilateral (*)
- When index equals content (*)
- Why a Diameter Is the Longest Chord?
- Wittenbauer's Parallelogram (*)
- Wundt Illusion (*)
- Wundt Block Illusion (*)
- Y. Sawayama's Lemma (*)
- Y. Sawayama's Theorem
- Zaslavsky's theorem (*)
- Zöllner's Illusion (*)
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