Metric Relations in a Right Triangle, Theorems and Problems

Lines interwoven,
Right triangle's metric dance,
Pythagorean ties.
Hypotenuse and legs meet,
Proportions sing in embrace.

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

The General Extension to Pythagoras' Theorem.

The Pythagorean Curiosity Triangles and squares, fifteen conclusions.

Word Cloud of Pythagorean Theorem

Einstein and Pythagoras theorem proof
 

Geoboard for iPad Pythagorean Theorem Proof by Leonardo.

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.

Geoboard for iPad Pythagorean Theorem Proof, Squares, (Problem 540).

Pythagorean spiral or Square Root Spiral or Spiral of Theodorus.

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

Geometry Problem 1603: One square. One circle. One secant from point C. A square inside a circle reveals a geometric mystery. Are you ready to solve it?

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

Geometry Problem 1598: Inradius Ratio with Median BM in a Right-Angled Triangle: A STEM Geometry Challenge

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

Geometry Problem 1596: Triangle Projections: Unraveling the DE Projection with Equal Angles!: A STEM Geometry Challenge

Geometry Problem 1589: Unlock the Mystery: Prove the Ratio of BD to OD is Square Root of Two in a 90° Circular Sector!.

Geometry Problem 1588: Unlocking the Tangent: Prove the Fifth of the Radius in a Sector with Semicircle.

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

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

Geometry Problem 1539 Demystified: Unraveling the Lengths in an Isosceles Triangle with Altitude and Tangent Secrets!.

Geometry Problem 1502.
Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral.

Geometry Problem 1022.
Circular Sector of 90 Degrees, Squares, Metric Relations.

Geometry Problem 1013.
Square, Line through the Center, Perpendicular, Distance, Metric Relations.

Geometry Problem 973.
Three Parallel, Square, Vertices, Distance Between Two Parallel Lines, Side, Metric Relations.

Geometry Problem 787
Right Triangle, Inradius, 18 Equal Circles, Line Tangent, Legs, Hypotenuse, Ratio, Metric Relations.

Geometry Problem 786
Right Triangle, Altitude, Hypotenuse, Cevian, Perpendicular, Ratio, Metric relations.

Geometry Problem 760
Isosceles Triangle, Circle, Diameter, Altitude, Chord, Metric relations.

Geometry Problem 700.
Equilateral Triangle, Circle, Circular Segment, Midpoint of a side, Metric Relations.

Geometry Problem 699.
Equilateral Triangle, Circle, Circular Segment, Midpoint, Metric Relations.

Geometry Problem 676.
Circles, Diameter, Tangent, Chord, Metric Relations.

Geometry Problem 651.
Archimedes Book of Lemmas: Proposition 11
Perpendicular chords, Arcs, Radius.

Geometry Problem 650.
Tangent Circles, Diameter, Radius, Chord, Perpendicular.

Geometry Problem 608
Square with Right Triangles, Metric Relations, Congruence, Measurement.

Geometry Problem 607
Intersecting Circles, Triangles, Metric Relations, Measurement.

Geometry Problem 605
Post a comment or solution
Circle, Square, Chord, Perpendicular, Circular Segment, Metric Relations.

Problem 582.
Quadrilateral, Diagonals, Incenters, Metric Relations.

Problem 571.
Circle, Diameter, Chord, Perpendicular, Measurement.

Problem 544.
Right Triangle, Altitude, Median, Equal angles, Measure.

Problem 530.
Cyclic Quadrilateral, Diagonal, Diameter, Chord, Perpendicular, Congruence.

Problem 534.
Right Triangle, Squares, Perpendicular, Dissection, Areas.

Problem 528.
Triangle, Medians, Perpendicular, Measurement.

Problem 522.
Right Triangle, Circle, Diameter, Tangent.

Problem 491.
Square, Right Triangle, Incircle, Inscribed Circle, Radius.

Problem 485.
Square, Side, Circles, Tangent, Radius, Measurement.

Proposed Problem 461.
Three Circles, Tangent, Right Angle, Center, Distance, Measurement.

Proposed Problem 460.
Circle, Tangent, Perpendicular, Radius, Distance, Measurement.

Proposed Problem 459.
Right triangle, Squares, Distance, Measurement.

Proposed Problem 458.
Square, Semicircle, Circular Sector, Internal Common Tangent, Measurement.

Proposed Problem 449.
Congruent Tangent Circles, Diameter, Sagitta, Arc, Perpendicular, Chord.

Proposed Problem 437.
Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.

Proposed Problem 436.
Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.

Proposed Problem 429.
Circumscribed and Inscribed Regular Pentagon, Perpendicular, Measurement.
Pythagoras

Proposed Problem 399.
Right triangle, Midpoints, Distance, Pythagorean theorem, Congruence.

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

Proposed Problem 336.
Two equal circles, a Common Tangent and a Square.

Proposed Problem 317.
Right triangle and Inscribed Squares.

Proposed Problem 316.
Circular segments and Inscribed Squares.

Proposed Problem 314.
Tangent circles, Common external tangent line, Geometric Mean.

Proposed Problem 303.
Triangle, Angle bisector of 90 degrees.

Proposed Problem 284.
Circular Sector 90 degrees, Semicircles, Tangent, Radius.

Proposed Problem 278.
Tangent Circles, Common External Tangent, Chord.

Proposed Problem 277.
Tangent Circles, Common External Tangent.

Proposed Problem 276.
Square, 90 degree Arcs, Circle, Radius.

Proposed Problem 275.
Right Triangle, Circumcircle, Sagitta, Inradius.

Proposed Problem 271.
Tangent Circles, the Cube of the Common external tangent.

Proposed Problem 270.
Tangent Circles, Common external tangent, Fractional exponents.

Proposed Problem 269.
Right Triangle, Altitude, Perpendicular, Projection.

Proposed Problem 268.
Right Triangle, Catheti and Altitude.

Proposed Problem 267.
Right Triangle, Product of Catheti.

Proposed Problem 266.
Right Triangle, Altitude, Geometric Mean.

Proposed Problem 265.
Right Triangle, Pythagorean Theorem.

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

Proposed Problem 239. Square, Midpoints, Lines, Congruence, Pythagoras.

Proposed Problem 207. Right Triangle, Hypotenuse, Inradius, Exradius relative to the hypotenuse.

Proposed Problem 204. Right Triangle, Incircle, Excircles, Inradius, Exradii.

Proposed Problem 203. Right Triangle, Excircles, Exradii, Hypotenuse.

Proposed Problem 202. Right Triangle, Incirle, Excircles relatives to catheti, Points of Tangency, Exradius, Semiperimeter.

Proposed Problem 201. Right Triangle, Excircles, Points of Tangency, Exradius, Semiperimeter.

Proposed Problem 200. Right Triangle, Incircle, Excircles, Points of Tangency, Inradius.

Proposed Problem 190. Tangent circles, Tangent chord, Perpendicular, Distance, Intersection of a Line with a Circle, Diameter.

Proposed Problem 183. Right Triangle, Hypotenuse Trisection Points, Squares of the Distances.