Online Geometry: Similarity, Ratios, Proportions. Theorems and Problems

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

Right Triangles, Formulas and Facts
Pythagorean theorem, Similarity, Geometric mean, hypotenuse, legs, semiperimeter, altitude, median, similar shapes areas, centers, orthocenter, incenter, centroid, circumcenter, radius, inradius, circumradius, exradius, circles, incircle, circumcircle, excircle.

Euclid's Elements, Book X, Lemma for Proposition 33
One page visual illustration.

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

Albert Einstein and the Pythagorean theorem.
A preteen Albert Einstein devised a new proof of the Pythagorean theorem.

Clawson Point.
Orthic and Extangents Triangles, Concurrent lines.

Geometry Problem 865
Parallelogram, Diagonal, Midpoint, Side, Triangle, Parallel, Similarity.

Geometry Problem 864
Parallelogram, Diagonal, Congruence, Similarity, Metric Relations.

Geometry Problem 847
van Lamoen Circle, Triangle, Medians, Centroid, Six Circumcenters, Concyclic Points.

Geometry Problem 846
Triangle, Medians, Centroid, Four Circumcenters, Concyclic Points, Cyclic Quadrilateral, Similarity.

Geometry Problem 845
Triangle, Medians, Centroid, Four Circumcenters, Perpendicular, Congruence, Similarity.

Geometry Problem 844
Triangle, Medians, Centroid, Four Circumcenters, Perpendicular, Congruence, Similarity.

Geometry Problem 843
Triangle, Medians, Centroid, Four Circumcenters, Perpendicular, Parallel, Congruence, Similarity.

Geometry Problem 842
Triangle, Medians, Centroid, Four Circumcenters, Perpendicular, Congruence, Similarity.

Geometry Problem 841
Triangle, Medians, Centroid, Circumcenters, Perpendicular, Congruence, Similarity.

Geometry Problem 840
Parallelogram, Perpendicular, Diagonal, Metric Relations, Similarity.

Geometry Problem 839
Problem 839: Parallelogram, Perpendicular, Diagonal, Metric Relations.

Geometry Problem 836
Parallelogram, Perpendicular, Diagonal, Similarity, Metric Relations.

Geometry Problem 834
Parallelogram, Diagonal, Similarity, Geometric Mean, Mean Proportional.

Geometry Problem 833
Parallelogram, Diagonal, Similarity, Proportion.

Geometry Problem 806
Triangle, Points on sides, Circumcenter, Similarity.

Geometry Problem 797
Parallelogram, Angle Bisector, Isosceles and Similar Triangles.

Geometry Problem 772
Similar triangles with a Common Vertex, Equal Angles, Midpoints, Metric relations.

Geometry Problem 751.
Parallelogram, Congruence, Intersecting Lines, Angle Bisector.

Geometry Problem 750.
Complete Quadrilateral, Triangle, Diagonal, Parallel, Similarity, Metric Relations.

Geometry Problem 747.
Trapezoid, Diagonals, Parallel, Bases, Midpoint, Similarity, Harmonic Mean, Metric Relations.

Geometry Problem 746.
Triangle, Cevian, Parallel, Similarity, Harmonic Mean, Metric Relations.

Geometry Problem 710.
Parallelogram, Midpoint, Diagonal, Collinear Points, Metric Relations, Triangle, Similarity, Congruence.

Geometry Problem 679.
Triangle, Circle, Median, Proportion.

Geometry Problem 677.
Parallelogram, Midpoint, Diagonal, Metric Relations.

Geometry Problem 626
Triangle, Distance from the Incenter to an Excenter.

Geometry Problem 625
Triangle, Parallel Lines, Congruence, Angle Bisector, Similarity.

Proposed Problem 434.
Quadrilateral, Transversal, Ratio, Measurement, Similarity.

Proposed Problem 433.
Quadrilateral, Triangle, Area, Proportion, Measurement, Similarity.

Proposed Problem 432.
Trapezoid, Parallel, Measurement, Similarity, Transversal.

Proposed Problem 431.
Quadrilateral, Midpoints of Diagonals, Transversal.

Proposed Problem 430.
Circumscribed and Inscribed Regular Pentagon, Perpendicular, Areas, Similarity.

Proposed Problem 367.
Square, Angle, 45 degrees, Pythagoras, Congruence, Similarity.

Proposed Problem 265.
Right Triangle, Pythagorean Theorem, Similarity.

Proposed Problem 264.
Right Triangle, Altitude, Leg projection, Hypotenuse, Similarity, Geometric mean.

Mascheroni Construction with compass alone. Midpoint of a segment, Similarity.

Stewart Theorem Triangle, sides and a cevian.

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

Heron's Formula with medians 

Gergonne Point Theorem. Concurrency. Interactive proof with animation.

Key concept: Ceva's Theorem.
 

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

Puzzle of the Ceva's Theorem: 48 classic piece.

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

Gergonne's Theorem: Triangle and Concurrent Cevians.

Van Aubel's Theorem II: Triangle and Cevians.
 

Proposed Problem 64: Triangle, Incircle, Transversal.

Triangle with the bisectors of the exterior angles. Collinearity. Key concept: Menelaus Theorem.
 

Ptolemy's Theorem.
 

Cyclic Quadrilateral: Ratio of the Diagonals
 

Casey's Theorem. Generalized Ptolemy's Theorem.

Pascal's Mystic Hexagram Theorem Proof

Eyeball Theorem: Animated Angle to Geometry Study.

Butterfly Theorem Proof with animation.

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

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.

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.

Blanchet Theorem

Lemoine Theorem

Nagel Point Theorem. Proof.
 

Pentagons & Pentagrams. I found, exploring with dynamic geometry software, new facts about pentagons and pentagrams. Key concept: Menelaus Theorem.

Interactive Hinged tessellation. Square tessellation - Dynamic Geometry.
Please be patient while the applet loads on your computer. This Java applet requires Java 1.3 or higher.

Problem 99: Circle Area, General Extension to Pythagoras' Theorem.

Proposed Problem 62: Square Diagonal, Inscribed Circle.

Proposed Problem 40.
Triangle, Incenter, Excenter, Angles 80, 40, Distances.

Proposed Problem 40. Geometry Help.
 

Sangaku Proposed Problem 41. "Mickey Mouse."
Three circles, tangents, radii.

Proposed Problem 63: Heptagon Regular, Side and Diagonals.

Morley's Triangle & Center: with interactive animation and manipulation.

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

The Pythagorean Curiosity Triangles and squares, fifteen conclusions.

Complete Quadrilateral: Ortholine-Steiner Line.
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.

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.
 
Proposition 4, Arbelos,

Archimedes' Book of Lemmas.
Proposition 5, Arbelos, Archimedean Twins.

Archimedes' Book of Lemmas.
Proposition 6, Arbelos,

Archimedes' Book of Lemmas.
Proposition 7, Similarity of Areas.

Archimedes' Book of Lemmas.
Proposition 10, Circle, Tangents, Chord, Parallel

Archimedes' Book of Lemmas.
Proposition 11, Perpendicular chords

I have used Geometry Expressions, the world's first Interactive Symbolic Geometry System, to visualize the Archimedean Twins and check out a variety of conjectures.

Geometry Problem 624
Triangle, Incenter, Excenter, Incircle, Excircle, Similarity.

Proposed Problem 131. Van Aubel Theorem, Concurrent Cevians, Sum of Ratios.

Proposed Problem 130. Triangle, Concurrent Cevians, Sum of Ratios.

Proposed Problem 129. Triangle, Concurrent Cevians, Sum of Ratios.

Proposed Problem 128. Incenter of a Triangle, Angle Bisectors, Sum of Ratios.

Proposed Problem 127. Centroid and Incenter of a Triangle, Parallel, Proportions.

Proposed Problem 126. Incenter of Triangle, Angle Bisector, Proportions.

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

PProposed Problem 125. Area of Triangle, Star, Trisection of Sides.

PProposed Problem 124. Area of triangle, Similarity, Trisection of Sides.

Proposed Problem 123. Area of triangle, Similarity, Trisection of Sides.

Proposed Problem 122. Marion Walter's Theorem. Area of triangle and Hexagon, Trisection of Sides.

Proposed Problem 121. Similarity and Area of triangle, Trisection of Sides.

Proposed Problem 103.Equilateral Triangle Area, Interior Point, Heron's Formula.

Proposed Problem 79: Triangle. Similarity, Altitudes, Orthocenter, Incircles, Inradii.

Proposed Problem 70: Squares inscribed in a triangle, Similarity.

Proposed Problem 69: Square inscribed in a triangle, Similarity.

Proposed Problem 68: Triangle, Incircle, Inradius, Tangent, Similarity.

Proposed Problem 36.
Right Triangle, altitude, incircles and inradii.

Proposed Problem 24
Right Triangle, altitude, incircles and inradii.

Proposed Problem 22
Right Triangle, altitudes, incircles and inradii.

Proposed problem 20
Right Triangle, incircles and inradii.

Proposed Problem 66: Triangle, Excircle, Tangents, Geometric Mean.
 

Proposed Problem 61: Triangle, Trisection of Sides

Archimedes Proposition 7. Circles and Squares

Lunes of Hippocrates 4.

Lunes of Hippocrates 3
 

Lunes of Hippocrates 2
 

Lunes of Hippocrates 1
 

Pythagoras and Circle Area.

Sangaku Geometry Theorem: Three circles and a tangent line.

Proposed Problem 292.
Triangle, Parallel, Congruence, Similarity.

Proposed Problem 291.
Triangle, Circle, Circumradius, Perpendicular.

Proposed Problem 98. Quadrilateral Area, Centroid, Similarity.

Proposed Problem 97. Similar Triangles, Areas.

Proposed Problem 96. Similar Triangles, Incenters, Parallelogram.

Proposed Problem 95. Similar Triangles, Inradii, Parallel.

Proposed Problem 94. Similar Triangles, Circumcircles, Circumradii.

Proposed Problem 93. Similar Triangles, Circumcircles, Parallelogram.

Proposed Problem 92. Similar Triangles, Circumcircles, Circumradii, Parallel.

Proposed Problem 91. Similar Triangles, Altitude, Parallel.

Triangle and Squares 17