Online Geometry: Classical Theorems of Euclidean Geometry - Table of Content

Pythagorean Theorem.

Euclid's Elements Book II, Proposition 12: Law of Cosines.

Euclid's Elements Book II, Proposition 13: Law of Cosines.

Median length, Apollonius' Theorem

The significance of the Pythagorean theorem  by Jacob Bronowski.
Pythagorean Theorem, 47th Proposition of Euclid's Book I.

Ceva's Theorem. Concurrency. Interactive proof with animation. Key concept: Menelaus Theorem. 

Menelaus' Theorem. Interactive proof with animation and key concepts..

van Aubel's Theorem. Quadrilateral with Squares. Proof with animation.

Heron's Formula.
Key facts and a purely geometric step-by-step proof.
 

Euclid's Elements Book I, 23 Definitions. One-page visual illustration.
Euclid's Elements Book. Index

Euclid's Elements Book VI, Proposition 3: Angle Bisector Theorem

Euclid's Elements, Book XIII, Proposition 10 One page visual illustration.

Ptolemy's Theorem.
 

Brianchon's Theorem in a Circumscribed Hexagon.

Brianchon Corollary, Circumscribed Hexagon, Concurrency lines.

Carnot's Theorem in an Acute Triangle.

Carnot's Theorem in an Obtuse Triangle.

Clifford's Circle Chain Theorems. This is a step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression.
 

Cyclic Quadrilateral: Ratio of the Diagonals
 

Nine-Point Center, Nine-Point Circle, Euler Line.
Interactive illustration.

Soddy Circles and Descartes Theorem.
Three tangent circles, Inscribed and Circumscribed Circles, Radii.

Euler's Problem, Problem 155. Distance between the Incenter to the Circumcenter.

The Simson Line, Theorems and Problems - Index.

Casey's Theorem. Generalized Ptolemy's Theorem.

Brahmagupta's Formula  Area of a cyclic quadrilateral.

Brahmagupta's Theorem Cyclic quadrilateral.

Platonic Solids, Interactive animation.
HTML5 Animation for iPad and Nexus
Flash Animation.

Theaetetus' Theorem, Platonic Solids, Interactive animation

Euler's Polyhedron Theorem

Pascal's Mystic Hexagram Theorem Proof

Pappus Theorem. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.

Apollonius' Tangency Problem for Three Circles Illustration with animation and sound.

Feuerbach Point Theorems
 

Feuerbach Points and Nine-Point Circle with interactive animation, manipulation, and step-by-step construction.

Angle between two Simson Lines. Proof with animation.
 

Simson Line. A proof of Simson line with animation.

Interactive Simson Line. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.

Lune of Hippocrates Index 

Lune of Hippocrates 4: Circle Areas and Right 

Hippocrates and Squaring the Circle
 

Stewart Theorem Triangle and a cevian.

Viviani's Theorem, Problem 221. Viviani's theorem, Equilateral triangle, Interior point, Distances.

Eyeball Theorem: Animated Angle to Geometry Study.

Blanchet Theorem

Sawayama -Thebault's theorem

Routh's Theorem - Index
Triangle, Cevians, Area, Ratio.
 

Parallelogram with Squares theorem Thébault's Theorem.
 

Johnson's Theorem, Intersecting circles.
HTML5 Animation
Adobe Flash Animation

Varignon and Wittenbauer theorems. Quadrilateral: midpoints and trisection points of the edges.

Bottema's Theorem:
Triangle and Squares with Interactive Geometry Software
Step-by-Step construction, Manipulation, and animation.

Morley's Theorem. Introduction with animation. Triangle + Trisectors = Equilateral triangle.

Monge & d'Alembert Three Circles Theorem II with Dynamic Geometry You can alter the geometric construction dynamically in order to test and prove (or disproved) conjectures and gain mathematical insight that is less readily available with static drawings by hand. Requires Java Plug-in 1.3 or higher. Please be patient while the applet loads on your computer. If you are using a dial-up connection, it may take a few minutes but is well worth the wait. Cabri, GSP, Cinderella, C.a.R.

Monge & d'Alembert Three Circles Theorem I with Dynamic Geometry You can alter the geometric construction dynamically in order to test and prove (or disproved) conjectures and gain mathematical insight that is less readily available with static drawings by hand. Requires Java Plug-in 1.3 or higher. Please be patient while the applet loads on your computer. If you are using a dial-up connection, it may take a few minutes but is well worth the wait. Cabri, GSP, Cinderella, C.a.R.

Miquel's Pentagram with Dynamic Geometry. You can alter the pentagram dynamically in order to test and prove (or disproved) conjectures and gain mathematical insight that is less readily available with static drawings by hand. Requires Java Plug-in 1.3 or higher. Please be patient while the applet loads on your computer. If you are using a dial-up connection, it may take a few minutes but is well worth the wait. Cabri, GSP, Cinderella, C.a.R.

Gergonne Point Theorem. Concurrency. Interactive proof with animation.

Key concept: Ceva's Theorem.
 

Nagel Point Theorem. Proof.
 

Napoleon's theorems and problems, Index.

Napoleon's Theorem. A purely geometric proof. It uses the Fermat point to prove Napoleon without transformations.
 

Interactive Gergonne Line and Nobbs Points. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.

Newton's Theorem, Newton-Gauss Line: Complete quadrilateral theorem. Using TracenPoche Dynamic Geometry Software, Online Step-by-Step construction, manipulation, and animation.

Newton's Theorem: Newton's Line. Circumscribed quadrilateral, midpoints of diagonals, center of the circle inscribed.

Schiffler Point: Four Euler Lines with interactive animation and manipulation.

Euler and his beautiful and extraordinary formula that links the 5 fundamental constants in Mathematics, namely, e, i, Pi , 1 and 0, together!
Adobe Flash Animation
HTML5 Animation for iPad and Nexus

Archimedes' Book of Lemmas. Exercise your brain. Archimedes wrote the "Book of Lemmas" more than 2200 years ago. Solve these 15 high school level problems and lift up your geometry skills.

Equal Incircles Theorem.  Interactive presentation.

Seven Circles Theorem.
Step by Step illustration of this beautiful theorem.

Second Ajima-Malfatti Point
Interactive illustration.

First Ajima-Malfatti Point
Animated illustration.

Lemoine Theorem

Fermat's Last Theorem
Central Michigan University, CMU, gets chance at math challenge

Adams' Circle Theorem
 

Langley's Problem Adventitious angles. 20° isosceles triangle with animation.
 

Mascheroni Construction with compass alone. Midpoint of a segment.

Geometry Problem 800: de Gua's Theorem
Pythagorean theorem in 3-D, Tetrahedron, Cubic Vertex, Triangular Pyramid, Apex, Height, Right Triangle Area, Base Area, Projected Area.

Fagnano's Problem
Inscribed Triangle with the Minimum Perimeter.

Isogonic-Jacobi Theorem: Using TracenPoche Dynamic Geometry Software

Taylor Circle Theorem Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation.

Marion Walter's Theorem and Related Topics: Index

Marion Walter's Theorem: Triangle and Hexagon areas:
Using TracenPoche Dynamic Geometry Software

Eight-Point Circle Theorem
Step-by-Step construction, Manipulation, and animation.

Complete Quadrilateral: Ortholine-Steiner Line.
Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation

Four Circles Theorem Using Interactive Dynamic Software
Step-by-Step construction, Manipulation, and animation.

Three Circles Theorem Using TracenPoche Dynamic Software

Sangaku Geometry Theorem: Three circles and a tangent line.

Sangaku Problem (An Old Japanese Theorem). Inradii, Carnot's theorem.

Van Aubel's Theorem II: Triangle and Cevians.
 

Kurschak's Tile and Theorem. Jozsef Kurschak (Hungary, 1864-1933) An elegant and a purely geometric way of finding the area of a regular dodecagon.
 

The Bevan Point The circumcenter of the excentral triangle. Illustration with animation.

Related Geometry Topics:
Computational, Manifold, Topology, String theory, Coordinate, Analytical, Trigonometry, Projective, Algebraic, Differential, Symplectic, Lie theory, Fractal, Dimension theory, Computer graphics, Atiyah, Gromov, Perelman.

Butterfly Theorem Proof with animation.