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 foursided 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 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.
. 

Geometry
Problem 1527: Discovering the Hidden Angle: Solving the Puzzle of Two Intersecting Circles.
. 

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, Stepbystep Illustration. 

Geometry
Problem 1454.
Intersecting Circles, Perpendicular Lines, Cyclic Quadrilateral, Collinear
Points,
iPad. Stepbystep illustration using GeoGebra. 

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

Dynamic Geometry 1452.
Japanese Theorem, Sangaku,
Cyclic Quadrilateral, Incenter, Rectangle, Inradius. Stepbystep
illustration using GeoGebra. 

Dynamic Geometry 1449.
Salmon Line. Stepbystep
illustration using GeoGebra. 

Dynamic Geometry 1448.
Simson Line. Stepbystep
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, NinePoint 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.
StepbyStep 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.
StepbyStep 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 Plugin 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. 

EightPoint Circle Theorem
StepbyStep construction, Manipulation, and animation. 

Complete Quadrilateral: OrtholineSteiner Line.
Using TracenPoche Dynamic Software
StepbyStep construction, Manipulation, and animation 
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