Master Geometry: Dive into Cyclic Quadrilateral Theorems and Problems

Cyclic quadrilateral,
Four vertices dance with grace,
Circle's loving hold.
Angles, sides harmonize,
Geometry's masterpiece unfold.

A cyclic quadrilateral is a four-sided polygon whose vertices lie on a circle. It has important properties that can be used to solve mathematical problems and has practical applications in fields such as engineering, physics, and architecture. One useful property is Ptolemy's theorem, which relates the diagonals and sides of a cyclic quadrilateral.

Ptolemy's Theorems and Problems - Index.

Brahmagupta Theorem and Problems - Index
Brahmagupta (598–668) was an Indian mathematician and astronomer who discovered a neat formula for the area of a cyclic quadrilateral.

Brahmagupta's Theorem Cyclic quadrilateral.

Cyclic Quadrilateral with Perpendicular Diagonals.

Geometry Problem 1583: Prove Lines CK and BL are Parallel.

Geometry Problem 1582: Prove That Angles AFD and AEF Are Equal in This Secant Problem.

Geometry Problem 1581: Prove the Angle Bisection in a Cyclic Quadrilateral.

Geometry Problem 1580: 60-Degree Triangle Challenge: Uncover AD Using Excenter and Circumcircle Clues.

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

Geometry Problem 1574: Triangle with Three Circles through a Point and the Concyclicity of Six Intersection Points

Geometry Problem 1573: Two circles are externally tangent at point C. Given specific secants and tangents, find the angle at point K

Geometry Problem 1568: Concyclicity of Points B, D , H, J in a Triangle ABC with an Incircle and a Tangent Circle.

Geometry Problem 1566: Demonstrate AC Equals AE + CD in an Equilateral Triangle ABC Involving Cevians and 60 Degrees..

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

Geometry Problem 1551: Unraveling Angle Relations in Cyclic Quadrilaterals: Solving for Angle GEJ: A High School Level Investigation

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

Geometry Problem 1533: Discovering Relationships between Angles and Lines in an Exterior Right Triangle of a Square - 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 1500.
Circle, Perpendicular, Tangent, Secant, Cyclic Quadrilateral, Parallel, Similarity, Measurement.

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

Geometry Problem 1491.
Cyclic Quadrilateral, Diagonal, Incircle, Angle, Measurement.

Dynamic Geometry 1472.
Cyclic Quadrilateral, Perpendicular Diagonals, Rectangle, Step-by-step Illustration.

Geometry Problem 1454.
Intersecting Circles, Perpendicular Lines, Cyclic Quadrilateral, Collinear Points, iPad. Step-by-step illustration using GeoGebra.

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

Dynamic Geometry 1452.
Japanese Theorem, Sangaku, Cyclic Quadrilateral, Incenter, Rectangle, Inradius. Step-by-step illustration using GeoGebra.

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

Dynamic Geometry 1448.
Simson Line. Step-by-step illustration using GeoGebra.

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

Geometry Problem 1399.
Triangle, Circumcircle, Perpendiculars, Angles, Congruence, Cyclic Quadrilateral.

Geometry Problem 1392.
Two Cyclic Quadrilateral, Cyclic Octagon, Circle.

Geometry Problem 1360.
Triangle, Nine-Point Circle, Feuerbach's Circle, Euler's Circle, Cyclic Quadrilateral, Concyclic Points, Sketch, iPad Apps.

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

Geometry Problem 1311.
Inscribed Triangle, Circle, Circumcircle, Concyclic Points, Measurement.

Geometry Problem 1306.
Cyclic, Inscribed Quadrilateral, Circle, Tangent, Diagonals, Collinear Points.

Geometry Problem 1271.
Triangle, Incircle, Incenter, Excircle, Excenter, Escribed Circle, Tangency Points, Six Concyclic Points.

Geometry Problem 1268.
Triangle, Exterior Angle Bisector, Circumcircle, Circle, Chord, Parallel, Concyclic Points, Cyclic Quadrilateral.

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

Geometry Problem 1258.
Cyclic Quadrilateral, Concyclic Points, Measurement.

Geometry Problem 1254.
Circle, Arc, Chord, Midpoint, Cyclic Quadrilateral, Concyclic Points.

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

Geometry Problem 1249.
Cyclic Quadrilateral, Circle, Triangle, Circumcircle, Angle Bisector, Parallel Lines, Area.

Geometry Problem 1245.
Cyclic or Inscribed Quadrilateral, Circle, Arc, Midpoint, Measurement. Mobile Apps.

Geometry Problem 1241.
Quadrilateral, Four Exterior Angle Bisectors, Cyclic Quadrilateral, Circle. Mobile Apps.

Geometry Problem 1234
Quadrilateral, 60 Degrees, Midpoint, Congruence, Cyclic Quadrilateral, Concyclic Points.

Geometry Problem 1227
Triangle, Circumcircle, Angle Bisector, Concyclic Points, Cyclic Quadrilateral.

Geometry Problem 1217
Triangle, Circle, Excenter, Incenter, Angle Bisector, Cyclic Quadrilateral, Circumcircle, Tangent Line.

Geometry Problem 1213
Triangle, Circumcircle, Altitudes, Cyclic Quadrilateral, Congruence.

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

Geometry Problem 1176
Cyclic Quadrilateral, Diagonals, Six Diameters, Circles, Collinear Points, Concurrent Lines.

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

Geometry Problem 1163.
Isosceles Triangle, Congruence, Double Angle, Circle, Concyclic Points, Circumcenter, Perpendicular, Angle Bisector.

Geometry Problem 1142.
Cyclic Quadrilateral, Isogonal Lines, Circle, Center, Radius, Perpendicular Bisector, Circumcenter.

Geometry Problem 1130.
Cyclic Quadrilateral, Diagonals, Angle Bisector, Congruent Angles.

Geometry Problem 1118.
Right Triangle, Angle Trisection, Concyclic Points, Cyclic Quadrilateral.

Geometry Problem 1113.
Triangle, Perpendicular Bisector, 90 Degrees, Circle, Concyclic Points, Cyclic Quadrilateral.

Geometry Problem 756: Four Circles Theorem
Using TracenPoche Interactive Dynamic Software.
Step-by-Step construction, Manipulation, and animation.

iPad Apps: Geometry Designer Illustration. Four Circles Theorem.
Interactive Geometry Software for iPad, iPhone, iPod.

Ptolemy's Theorem.
Cyclic quadrilateral 

Cyclic Quadrilateral: Ratio of the Diagonals
 

Brahmagupta's Corollary Cyclic quadrilateral.

Brahmagupta's Formula  Area of a cyclic quadrilateral.

Brahmagupta Formula Extension Area of any quadrilateral.

Angle between two Simson Lines. Proof with animation.
 

Simson Line. A proof of Simson line with animation.

Interactive Simson Line. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.

Miquel's Pentagram Theorem Interactive proof with animation and key theorems.
Pentagons and Cyclic quadrilateral. 

Miquel's Pentagram with Dynamic Geometry. You can alter the pentagram dynamically in order to test and prove (or disproved) conjectures and gain mathematical insight that is less readily available with static drawings by hand. Requires Java Plug-in 1.3 or higher. Please be patient while the applet loads on your computer.

Sangaku Problem. The incenters of four triangles in a cyclic quadrilateral form a rectangle.

Eight-Point Circle Theorem
Step-by-Step construction, Manipulation, and animation.

Complete Quadrilateral: Ortholine-Steiner Line.
Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation

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