


Explore the Applications of Concyclic Points with These Engaging Theorems and Problems
Concyclic points are a set of points that lie on the same
circle. In other words, these points share a common
circumcenter and have the same distance from that center
point. The term "concyclic" comes from the Latin words "con"
meaning "together" and "cyclic" meaning "circle".
This concept is often used in geometry, particularly in problems
involving triangles or polygons.
To solve
problems involving concyclic points and geometry, it's helpful
to remember the following properties:

Any two points on a
circle determine an arc, and the measure of the inscribed
angle that subtends that arc is half the measure of the
central angle that intercepts that arc.

The opposite angles of a cyclic quadrilateral add up
to 180 degrees.

The angle between a tangent and a chord of
a circle is equal to the measure of the inscribed angle that
subtends the chord.
Using these properties, we can solve
many geometry problems involving concyclic points.


Ptolemy's Theorems and Problems  Index.
Concyclic Points 

Four Circles Theorem Using Interactive Dynamic Software
StepbyStep construction, Manipulation, and animation.
Concyclic Points, Cyclic 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. 

GeoGebra, Dynamic Geometry:
Nine Point Circle
Nine concyclic points.
HTML5 Animation for Tablets (iPad, Nexus..) 

Lester Circle Theorem.
Concyclic points. 

van Lamoen Circle  Index. 

Miquel's Pentagram Theorem
Interactive proof with animation and key theorems.
Pentagons and
Concyclic Points. 

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.
Concyclic Points, 

EightPoint Circle Theorem
StepbyStep construction, Manipulation, and animation
Concyclic Points. 

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

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

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

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

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. 

Dynamic Geometry 1480.
Japanese Theorem for Cyclic Polygon, Sangaku, Triangulation, Nonintersecting Diagonals, Sum of Inradii, Invariant, Stepbystep Illustration. GeoGebra,
iPad. 

Dynamic Geometry 1479.
Triangle, Circumcircle, Angle Bisector,
Perpendicular Bisector, Chord, Concyclic Points, Parallel Lines,, Stepbystep Illustration. GeoGebra,
iPad. 

Dynamic Geometry 1477.
Miquel's Pentagram Theorem, Pentagon, Triangle, Circumcircles, Concyclic Points, Stepbystep Illustration. GeoGebra,
iPad. 

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

Geometry Problem 1366.
Square, Angle Bisector, Perpendicular, Concyclic Points, Circle, Sketch, iPad Apps. 

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 1294.
Triangle, Altitudes, Four Squares, Center, Midpoint, Concyclic Points. 

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

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

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

Geometry Problem 1239
Intersecting Circles, Secant, Concyclic Points, Congruence. 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 1163.
Isosceles Triangle, Congruence, Double Angle, Circle, Concyclic Points, Circumcenter, Perpendicular, Angle Bisector. 

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
1085.
Intersecting Circles, Concyclic Points, Cyclic Quadrilateral. 

Geometry Problem
1052.
Triangle, Quadrilateral, Three Circumcircles, Circle, Concyclic Points, Cyclic Quadrilateral. 

Geometry Problem
1017.
Square, Center, Midpoints, Circle, Concyclic Points, Cyclic Quadrilateral. 

Geometry Problem
1003.
Intersecting Circle, Circumcircle, Chord, Secant, Triangle, 90 Degree, Angle, Perpendicular Lines. 

Geometry Problem
1002.
Triangle, Circumcircle, Cevian, Parallel Lines, Cyclic Quadrilateral, Concyclic Points. 

Geometry Problem 993.
Intersecting Circles, Secant, Tangent, Concyclic Points, Cyclic Quadrilateral, Perpendicular. 

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

Geometry Problem 950.
Intersecting Circles, Secant, Cyclic Quadrilateral, Concyclic Points. 

Geometry Problem 949.
Intersecting Circles, Secant, Cyclic Quadrilateral, Concyclic Points. 
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