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
Step-by-Step 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.
Step-by-Step 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 Plug-in 1.3 or higher. Concyclic Points,

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

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

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, Non-intersecting Diagonals, Sum of Inradii, Invariant, Step-by-step Illustration. GeoGebra, iPad.

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

Dynamic Geometry 1477.
Miquel's Pentagram Theorem, Pentagon, Triangle, Circumcircles, Concyclic Points, Step-by-step 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, 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 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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