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.

Four Books on Human Proportion by Albrecht Durer and Golden Rectangles.

Clawson Point.
Orthic and Extangents Triangles, Concurrent lines.

Four Books on Human Proportion, Body Parts by Albrecht Durer and Golden Rectangles.

Geometry Problem 1604: Explore this elegant problem involving two circles, a midpoint, a tangent, and an unknown segment

Geometry Problem 1600: When Cevians Cross: Find the Missing Area: A STEM Geometry Challenge

Geometry Problem 1599: Four Equal Squares, One Right Triangle: What’s the Ratio?: A STEM Geometry Challenge

Geometry Problem 1597: Circle-Triangle Intersection Problem: A STEM Geometry Challenge

Geometry Problem 1587: Uncover the Elegance of Triangle BGH's Area in Trapezoid ABCD.

Geometry Problem 1586: Unveiling a Unique Area Relationship in Three Equilateral Triangles

Geometry Problem 1585: Discover how a triangle's incircle reveals the secret of harmonious convergence

Geometry Problem 1584: Proving a Ratio Involving Incircle and Angle Bisector
Unveiling a Potentially Unexplored Geometric Relationshp.

Geometry Problem 1578: Find the Area of a Bicentric Quadrilateral with Perpendicular Extensions of Opposite Sides!.

Geometry Problem 1570: Calculating BE in Overlapping Squares: A High School and College Geometry Challenge.

Geometry Problem 1567: Finding Tangent Distances in a Circumscribed Isosceles Trapezoid.

Geometry Problem 1560: Trapezoid ABCD: Unlocking Angle Secrets at G and H.

Geometry Problem 1559: Proving BC Bisects Segment DE. This geometry problem challenges students at high school and college levels.

Geometry Problem 1555: Find Length of DE. The problem involves chords, tangent, circles, and intersections of line

Geometry Problem 1543: Calculating the Area of Quadrilateral ABED in a Square with a Side Length of 20 and an Intersecting Arc.

Geometry Problem 1547: Tangents' Dance: Exploring B-to-AC Distance in a Circle's Grasp

Geometry Problem 1542: Unraveling a Geometric Puzzle with a Circumscribed Right Triangle and Square to the Same Circle for School and College.

Geometry Problem 1541 Challenge: Unveiling BG Length in an Inscribed Quadrilateral with Harmonic Quaternary Insight for Academic Pursuit.

Geometry Problem 1540: Solving for the Length of Chord in a Circle: Analyzing Intersections and Given Values for Academic Pursuits.

Geometry Problem 1515 and a Thematic Poem.
Mastering Triangle Distance Calculation: Find the Distance from the Intersection of Medians to an Exterior Line. Difficulty Level: High School.

Geometry Problem 1512 and a Thematic Poem.
Finding the Length of a Segment in a Triangle with a Median and a Cevian with Given Ratio. Difficulty Level: High School.

Geometry Problem 1500.
Circle, Perpendicular, Tangent, Secant, Cyclic Quadrilateral, Parallel, Similarity, Measurement.

Geometry Problem 1497.
Triangle, Transversal, Same Ratio, Proportionality, Isosceles, Double Angle.

Geometry Problem 1457.
Altitudes, Circles, Similarity, Product of the Inradii Lengths, iPad. Step-by-step illustration using GeoGebra.

Geometry Problem 1456.
Altitudes, Orthic Triangle, Circumcircle, Parallel lines, Similarity, Area, iPad. Step-by-step illustration using GeoGebra.

Geometry Problem 1442.
Triangle, Altitudes, Orthocenter, Perpendicular, Congruent Angles, Similarity, Measurement, Art, Poster, Typography, iPad Apps.

Geometry Problem 1432.
Tangent Circles, Secant, Tangent Lines, Proportionality, Similarity, Poster Typography, iPad.

Geometry Problem 1424.
Triangle, Parallelogram, Trapezoid, Area, Diagonal, Measurement.

Geometry Problem 1423.
Triangle, Area, Three Equal Angles, Circumradius, Measurement.

Geometry Problem 1386.
Thabit ibn Qurra (826 -901) Theorem and more conclusions, generalized Pythagorean Theorem to any triangle.

Geometry Problem 1185
Similar Rectangles, Four Quadrilaterals, Sum of the Areas, Metric Relations.

Geometry Problem 1173
Triangle, Cevian, Angle, Congruence, Similarity.

Geometry Problem 1172
Cyclic Quadrilateral, Circle, Secant, Triangle, Similarity, Metric Relations.

Geometry Problem 1135.
Intersecting Circles, Secant, Parallel Line, 90 Degrees, Similar Triangles.

Geometry Problem 1133.
Circle, Chord, Tangent, Circumcircle, Similarity, Equal Product.

Geometry Problem 1126.
Equilateral Triangle, Circumcircle, Metric Relation, Harmonic Mean.

Geometry Problem 1119.
Quadrilateral, Similar Triangles, Midpoints, Equal Ratio, Angles.

Geometry Problem 1108.
Right Triangle, External Squares, Catheti, Congruence, Harmonic Mean.

Geometry Problem 1103.
Right Triangle, Incircle, Inscribed Circle, Radius, Geometric Mean, Sangaku, Japanese, Metric Relations.

Geometry Problem 1101.
Triangle, Parallel Lines, Circle, Incircle, Inradius, Radius, Angle.

Geometry Problem 978.
Parallelogram, Diagonal, Transversal, Similarity, Metric Relations.

Geometry Problem 960.
Quadrilateral, Trisection, Sides, Congruence, Similarity, Triangle.

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.