Master Diameters and Chords: Theorems, Properties, and Problems

Geometry Problem 1617: Tangency, Concyclic Points, and Metric Relations in a Triangle

Geometry Problem 1614: Semicircle, Inscribed Rectangles, Area Equivalence

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

Geometry Problem 1592: A Semicircle Challenge for Mental Clarity Involving Radii, Perpendiculars, Midpoint

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 1579: Involving perpendicular diameters, tangents, and triangle areas.

Geometry Problem 1569: Prove a Relationship: EG Equals the Incircle Diameter in Triangle ABC with Square CDEF.

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

Geometry Problem 1561: Triangle ABC, Circumcenter O, Orthocenter H, Parallel Line, and Angle Secrets.

Geometry Problem 1556: Right Triangle ABC and Inscribed Circle. The problem involves circle, chords, tangent, perpendicular lines, and congruence

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

Geometry Problem 1544: Challenge: Calculate the Area of a Triangle with Given Arc and Semicircle Intersections.

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

Geometry Problem 1532: Crack the Code of Geometry Problem 1532: How to Find the Angle in a Square with a Tangent Semicircle! - A High School Challenge.
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Geometry Problem 1531: Discover How to Calculate the Length of a Chord in a Circle with Diameter Intersection and an Angle between the Diameter and Chord - A High School Challenge.
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Geometry Problem 1527: Discovering the Hidden Angle: Solving the Puzzle of Two Intersecting Circles.
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Geometry Problem 1521 and a Thematic Poem.
Unlock the Secret to Finding the Measure of an Angle in a Triangle with Two Sides as Diameters of Circles.

Geometry Problem 1498.
Intersecting Circles, Diameter, Common Chord, Secant, Cyclic Quadrilateral, Concurrent Lines, Concyclic and Collinear Points.

Geometry Problem 1495.
Circle, Parallel Chords, 30 Degree Angle, Radius Squared.

Geometry Problem 1485.
Triangle, Orthocenter, Altitude, Circle, Diameter, Tangent, Measurement.

Dynamic Geometry 1479.
Triangle, Circumcircle, Angle Bisector, Perpendicular Bisector, Chord, Concyclic Points, Parallel Lines,, Step-by-step Illustration. GeoGebra, iPad.

Dynamic Geometry 1474.
Butterfly Theorem, Circle, Chords, Midpoints, Step-by-step Illustration.

Geometry Problem 1453.
Two Semicircles, Cyclic Quadrilateral, Concyclic Points, iPad.

Dynamic Geometry 1449.
Salmon Line. Step-by-step illustration using GeoGebra.

Geometry Problem 1439.
Rectangle, Diagonal, Perpendicular, Circles, Areas, Art, Poster, Typography, iPad Apps.

Geometry Problem 1435.
Circle, Diameter, Inscribed Circles, Circular Sector, Parallelogram, Parallel Lines, Tangency Points, Poster, Typography, iPad Apps.

Geometry Problem 1431.
Triangle, Circumcircle, Parallel Chord, Apollonius, LLC, Congruent Angles, Poster, Conformal Mapping.

Geometry Problem 1402.
Triangle, Three Tangent Circles, Parallel Diameter, Collinear Points.

Geometry Problem 1401.
Right Triangle with three circles on the sides, Isosceles, Diameter, Center, Tangent, Congruence.

Geometry Problem 1397.
Circle, Diameter, Tangent, Secant, 90 Degrees Angle, Perpendicular, Collinear Points.

Geometry Problem 1382.
Circle, Diameter, Radius, Perpendicular, Congruence.

Geometry Problem 1380.
Common Internal Tangents. Circles, Secant, Congruent Chords.

Geometry Problem 1379.
Common External Tangents. Circles, Secant, Congruent Chords.

Geometry Problem 1365.
Circle, Tangent, Secant, Chord, Parallel, Midpoint, Sketch, iPad Apps.

Geometry Problem 1355.
Triangle, Midpoint, Median, Circle, Chord, Equal Product of measure of segments.

Geometry Problem 1354.
Circle, Tangent Lines, Secant, Chord, Center, Parallel Lines.

Geometry Problem 1353.
Tangent Circles, Perpendicular Diameters, Congruence, 30 Degrees.

Problem 1343.
Triangle, Excircles, Circle, Tangent, Tangency Points, Chord, Perpendicular, 90 Degrees, Collinearity. Machu Picchu in the background.

Problem 1342.
Circle, Secant, Chord, Midpoint, Concyclic Points, Cyclic Quadrilateral.

Geometry Problem 1321.
Intersecting Circles, Diameter, Secant, Chord, Collinearity.

Geometry Problem 1281.
Triangle, Circle, Diameter, Tangent, Altitude, Congruent Angles.

Geometry Problem 1261.
Cyclic Quadrilateral, Circle, Diameter, Midpoint, Measurement.

Geometry Problem 1259.
Triangle, Circumcircle, Circle, Diameter, Parallel Lines.

Geometry Problem 1251.
Triangle, Circle, Diameter, Perpendicular, 90 Degrees.

Geometry Problem 1250.
Cyclic or Inscribed Quadrilateral, Circle, Diameter, Congruence.

Geometry Problem 1191
Circle, Chords, Diameter, Congruence, Midpoint, Collinearity, Bisector.

Geometry Problem 1190
Circle, Chords. Congruence.

Geometry Problem 1181
Cyclic Quadrilateral and Tangential Quadrilateral, Diameter as a Diagonal, Incenter, Circumcenter.

Geometry Problem 1151.
Circle, Diameter, Chord, Perpendicular, 90 Degrees, Metric Relations.

Geometry Problem 1090.
Circle, Diameters, Chord, Perpendicular, Metric Relations.

Geometry Problem 1084.
Circle, Chord, Radius, Diameter, Parallel lines, 45 Degrees, Metric Relations.

Geometry Problem 1083.
Semicircle, Diameter, Perpendicular, 90 Degrees, Tangent Line, Metric Relations.

Geometry Problem 1071.
Arbelos, Semicircles, Diameter, Circle, Perpendicular, Common Tangent, Parallel, Collinear Points, 90 Degrees, Midpoint.

Geometry Problem 1070.
Circle, Chord, Equilateral Triangle, Square, Rectangle, Area, Diameter, Perpendicular.

Geometry Problem 1058.
Triangle, Sum of the Areas, Altitude, Perpendicular, Circumcircle, Circle, Diameter.

Geometry Problem 1057.
Triangle, Area, Circle with Perpendicular Diameters, Tangent, Secant.

Geometry Problem 1048.
Circles, Tangent, Perpendicular, Diameter, Angle Bisector.

Geometry Problem 1033.
Triangle, Angle Obtuse, Circumcircle, Diameter, Area.

Geometry Problem 1019.
Circle, Diameter, Perpendicular, Chord, Tangent, Sum.

Geometry Problem 951.
Intersecting Circles, Chord, Diameter, Parallel, Cyclic Quadrilateral, Concyclic Points.

Geometry Problem 783
Triangle, Orthocenter, Circumcenter, Circumcircle, Diameter, Altitude, Midpoint.

Geometry Problem 782
Triangle, Orthocenter, Circumcenter, Circumcircle, Diameter, Altitude, Parallelogram.

Geometry Problem 778
Triangle, Distance from the Orthocenter to a Vertex, Circle, Circumradius, Side, Square.

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

Geometry Problem 713.
Tangent Circles, Diameter, Chord, Tangent, Incenter, Incircle, Triangle.

Geometry Problem 712.
Semicircle, Diameter, Tangent, Perpendicular, Angle Measure.

Geometry Problem 708.
Circle, Tangent, Intersecting, Lune, Area, Diameter.

Geometry Problem 705.
Intersecting Circles, Diameter, Angles, Measurement.

Geometry Problem 701.
Intersecting Circles, Diameter, Perpendicular, Angles, Congruence.

Geometry Problem 680.
Concentric Circles, Radii, Chords, Perpendicular, Metric Relations.

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

Geometry Problem 672.
Internally tangent circles, Chord, Tangent, Geometric Mean.

Geometry Problem 658.
Triangle, Altitudes, Diameter, Circle, Angles.

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

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

Problem 529.
Right Trapezoid, Circle, Diameter, Chord.

Problem 525.
Circles, Diameter, Chord, Tangent, Radius, Congruence, Measurement.

Problem 523.
Tangent Circles, Diameter, Chord, Perpendicular, Collinearity.

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

Problem 500.
Circle, Diameter, Chord, Perpendicular, Measure.

Problem 487.
Intersecting Circles, Area, Diameter and Chord, Parallel, 90 Degrees.

Problem 480.
Triangle, Circle, Center, Diameter, Chord, Altitude, Angle, Circumcenter.

Problem 478.
Triangle, Sides, Circumradius, Circumcenter, Circumcircle, Diameter, Chord.

Proposed Problem 456.
Three tangent circles, Center, Chord, Secant, Collinear, Diameters and Chords.

Proposed Problem 448.
Tangent Circles, Diameter, Perpendicular, Chord.

Proposed Problem 438.
Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement, Parallel lines.

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

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

Proposed Problem 373.
Square, Inscribed Circle, Diagonal, Perpendicular, Angle, Diameters and Chords.

Proposed Problem 364.
Obtuse triangle, Congruence, Circles, Diameter, Chord, Curvilinear triangle, Area.

Proposed Problem 333.
Circle inscribed in a semicircle, Perpendicular to the common tangent, Diameters and Chords.

Proposed Problem 329.
Triangle, Altitudes, Circle, Diameter, Concyclic points, Diameters and Chords.

Proposed Problem 310.
Circle Inscribed in a Semicircle, 45 Degrees Angle, Diameters and Chords.

Proposed Problem 294.
Right triangle, Circumcenter, Excenter, Hypotenuse, Perpendicular, Diameters and Chords.

Proposed Problem 272.
Tangent Circles, the Cube of the Common external tangent, Diameters and Chords.

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

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