# 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 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. 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. Eyeball Theorem: Animated Angle to Geometry Study. Butterfly Theorem Proof with animation. Nine-Point Center, Nine-Point Circle, Euler Line. Interactive illustration.

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