Geometric Challenge Geometry Problem 1617: The Concyclic Hexagon and the Power of Intersecting Circumcircles

Share your proof or solution in the comments below.
Target Audience: K-12, Honors Geometry, and College Mathematics Education.

Discover the hidden symmetry of intersecting systems in Problem 1617. This challenge moves away from simple tangency to explore a more complex configuration where two circles overlap, creating a remarkable "Concyclic Hexagon."
Explore the full theorem and interactive diagrams by clicking the illustration below.

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Proposed Solution
We invite students, teachers, and enthusiasts to share their proofs. This classic challenge can be approached using synthetic geometry.

How to contribute:
Post your step-by-step proof in the comments below. Feel free to:

• Describe your construction and properties applied (centroid/medians).
• Provide a link to a diagram (GeoGebra, Desmos, etc.) if you have one.

Be the first to submit a formal solution!

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Please use the box below to Enter your Comment or Solution. You can use plain text or provide links to your diagrams.

Geometry Problem 1574: Triangle with Three Circles through a Point and the Concyclicity

Challenging Geometry Puzzle: Problem 1574. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Share your solution by clicking 'Comment' below the post or entering your solution or comment in the 'Enter Comment' field and pressing 'Publish'.

Geometry Problem 1568: Prove Concyclicity of Points B, D , H, J in a Triangle ABC

Challenging Geometry Puzzle: Problem 1568. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Share your solution by clicking 'Comment' below the post or entering your solution or comment in the 'Enter Comment' field and pressing 'Publish'.

Geometry Problem 1530: Unlock the Secrets of Geometric Angles: Calculate the Measurement of an Angle in a Square and Rectangle Figure Today

Geometry Problem 1530. Post your solution in the comment box below.
Level: Mathematics Education, K-12 School, Honors Geometry, College.

Details: Click on the figure below.

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To post a solution to this problem click Comment underneath the post, or click into the line that says, “Enter Comment.” Type what you want to say and press Publish to post your solution.

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

Geometry Problem 1498. Post your solution in the comment box below.
Level: Mathematics Education, K-12 School, Honors Geometry, College.

Details: Click on the figure below.

More Details

To post a solution to this problem click Comment underneath the post, or click into the line that says, “Enter Comment.” Type what you want to say and press Publish to post your solution.

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

Interactive step-by-step animation using GeoGebra. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Interactive step-by-step animation using GeoGebra. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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Geometry Problem 1472: Cyclic Quadrilateral, Perpendicular Diagonals, Rectangle

Interactive step-by-step animation using GeoGebra. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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Geometry Problem 1453: Two Semicircles, Cyclic Quadrilateral, Concyclic Points

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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Typography of Geometry Problem 71: Cyclic Quadrilateral, Angles, Concyclic Points, iPad Apps

Geometric Art: Typography, iPad

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Typography of Geometry Problem 866: Cyclic Quadrilateral, Circle, Rectangle, Center, Congruence, 90 Degrees, iPad Apps

Geometric Art: Typography, iPad

Details: Click on the figure below.

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Geometry Problem 548 Triangle, Circle, Concyclic Points, iPad Apps, Typography

Geometric Art: Typography, iPad

Details: Click on the figure below.

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Geometry Problem 1342: Circle, Secant, Chord, Midpoint, Concyclic Points, Cyclic Quadrilateral

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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Geometry Problem 1294 Triangle, Altitudes, Four Squares, Center, Midpoint, Concyclic Points

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

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

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

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