GoGeometry.com (Problem Solutions): area

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Friday, September 25, 2026

Geometric Challenge
Right Triangle, Incircle, Circumscribed Square, Inradius, Area

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

In Problem 1624, we explore a right triangle and a square circumscribed about its incircle with one side resting on the hypotenuse. The challenge is to prove that the area of the square S and the inradius r satisfy the fundamental geometric mean identity S = 4r² = 2mn, where m and n are the hypotenuse segments adjacent to the square.
Explore the full theorem and illustrated diagram by clicking the image below.

Illustration of Geometry Problem 1624: Right Triangle, Incircle, Circumscribed Square, Inradius, Area

Click for additional details and full diagram.

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

Describe the theorems applied.
Share a link to your dynamic construction (GeoGebra, Desmos).

Be the first to submit a formal solution for this problem!

Ready to contribute?
Please use the box below to Enter your Comment or Solution. You can use plain text or provide links to your digital proofs.

Thursday, April 2, 2026

Geometric Challenge
Problem 1615 -Quadrilateral, Midpoints, and Area Invariance, Synthetic Geometry.

Geometry Challenge 1615: Share your proof or solution in the comments below.
Target Audience: K-12, Honors Geometry, and College Mathematics Education.

Explore the full theorem and interactive diagrams by clicking the illustration below.

Click for additional details.

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!

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

Saturday, March 14, 2026

Geometric Challenge
Problem 1614 - Area Equivalence in a Semicircle with Inscribed Rectangles, Synthetic Geometry.

Geometry Challenge 1614: Share your proof or solution in the comments below.
Target Audience: K-12, Honors Geometry, and College Mathematics Education.

Explore the full theorem and interactive diagrams by clicking the illustration below.

Click for additional details.

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!

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

Thursday, December 4, 2025

Geometry Problem 1610: Area of a Curvilinear Triangle in a Rhombus

Challenging Geometry Problem 1610. 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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