Online Geometry: Similarity, Ratios, Proportions. Theorems and Problems. Page 1 of 2

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 884: The Pythagorean Curiosity
Right Triangles and squares, fifteen conclusions.

Geometry Problem 877
Square, Right Triangle, Similarity, Metric Relations, Transversal.

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.