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.

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.

Geometry Problem 1502: Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral

Geometry Problem 1502. 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.

Geometry Problem 1436: Right Triangle, Altitude, Excircles, Excenters, Geometric Mean, Art, Poster

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

Details: Click on the figure below.

Zoom

Typography of Geometry Problem 1103: Right Triangle, Incircle, Inscribed Circle, Radius, Geometric Mean, iPad Apps

Geometric Art: Typography, iPad

Details: Click on the figure below.

Zoom

Typography of Geometry Problem 834: Parallelogram, Diagonal, Similarity, Geometric Mean, iPad Apps

Geometric Art: Typography, iPad

Details: Click on the figure below.

Zoom

Geometry Problem 1111: Right Triangle, External Squares, Cathetus, Angle Bisector, Area, Geometric Mean

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

Click the diagram below to enlarge it.

Zoom

Problem 1105: Triangle, Incircle, Excircle, Cevian, Tangent, Congruence, Geometric Mean

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

Click the diagram below to enlarge it.

Zoom

Problem 1103: Right Triangle, Incircle, Inscribed Circle, Radius, Geometric Mean, Sangaku, Japanese, Metric Relations

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

Click the diagram below to enlarge it.

Zoom

Problem 1095: Externally Tangent Circles, Common Tangent, Radius, Geometric Mean, Metric Relations

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

Click the diagram below to enlarge it.

Zoom

Geometry Problem 1020: Circle, Tangent, Chord, Perpendicular, Geometric Mean

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

Click the figure below to see the complete problem 1020.

Zoom

Geometry Problem 959. Triangle, Sides Ratio 4:1, Inradius, Exradius, Cevian, Mean Proportional, Geometric Mean

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 959.

Zoom

Problem 834: Parallelogram, Diagonal, Similarity, Geometric Mean, Mean Proportional

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 834.

Zoom

Geometry Problems 671-680: Triangle, Circle, Congruence, Geometric Mean, Tangent, 45 Degrees, Collinear Points, Simson Line, Concentric, Similarity

Ten Geometry Problems
Click the figure below to view the ten problems.

Zoom Problems 671 - 680

Level: High School, College geometry

Problem 672: Internally tangent circles, Chord, Tangent, Geometric Mean

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 672.

Zoom

Geometry Problems 311 - 320

Ten Geometry Problems
Click the figure below to see problems 311-320 about Circle, Semicircle, Chord, Diameter, Tangent, Cyclic quadrilateral, Geometric mean, Square, Right triangle, Altitude, Collinearity, Circumcircle, Incenter, Excenter.

Read more

Level: High School, SAT Prep, College geometry

Geometry Problems 261-270: Regular Polygon, Pentagon, Hexagon, Circle, Tangent, Similarity, Triangle, Hypotenuse, Catheti, Geometric Mean, Altitude,

Ten Geometry Problems
Click the figure below to see problems 261-270 about Regular Polygon, Pentagon, Hexagon, Circle, Distance, Triangle, Hypotenuse, Catheti, Similarity, Geometric Mean, Altitude, Perpendicular, Projection, Tangent, Fractional Exponent, Common External Tangent.

Read more

Level: High School, SAT Prep, College geometry

Geometry Problems 61-70: Triangle, Quadrilateral, Circle, Cyclic, Similarity

Geometry Problem
Click the figure below to see problems 61-70 about Triangle, Trisection, Transversal, Square, Heptagon, Diagonal, Circle, Incircle, Midpoint, Right triangle, Tangent, Geometric mean, Cyclic quadrilateral, Area, Angle, Similarity.


See also:
Problems 61-70

Level: High School, SAT Prep, College geometry

Problem 315: Three tangent circles, Common external tangent line, Geometric Mean

Proposed Problem
Click the figure below to see the complete problem 315.


See also:
Complete Problem 315
Level: High School, SAT Prep, College geometry

Geometric Mean, Theorems and Problems

Index
Click the figure below to see the Geometric Mean, Theorems and Problems Index


See more:
Geometric Mean, Theorems and Problems
Level: High School, SAT Prep, College geometry

Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square

Proposed Problem
Click the figure below to see the geometric illustration of Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square.


See more:
Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square
Level: High School, SAT Prep, College geometry