Geometric Challenge Problem 1622: Outer Hexagon Formed by External Squares on a Triangle

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

In Problem 1622, we explore a triangle with external squares on each side, forming an outer hexagon. The challenge is to prove that the sum of the squares of the six sides of the outer hexagon equals four times the sum of the squares of the original triangle's sides.
Explore the full theorem and illustrated diagram by clicking the image below.

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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!

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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 digital proofs.

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.

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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!

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.

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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Geometry Problem 1609: Radius of a Semicircle Tangent to Two Sides of a Triangle

Challenging Geometry Problem 1609. 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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Geometry Problem 1605: Triangle, Angle Bisector, and Circumcircle

Challenging Geometry Problem 1605. 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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Geometry Problem 1602: Find the Distance from the Semicircle’s Center to the Incenter

Challenging Geometry Puzzle: Problem 1602. 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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Geometry Problem 1600: When Cevians Cross: Find the Missing Area

Challenging Geometry Puzzle: Problem 1600. 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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Geometry Problem 1597: Circle-Triangle Intersection Problem

Challenging Geometry Puzzle: Problem 1597. 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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STEM-Powered Geometry Problems: Building Real-World Skills

Explore our new STEM Geometry Problem Collection—real-world challenges that blend geometry with science, tech, engineering, and math.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Dive In: Explore Now

Shape your problem-solving skills today!

Geometry Problem 1594: Intersecting Squares & Hidden Lengths

Challenging Geometry Puzzle: Problem 1594. 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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Geometry Problem 1592: A Semicircle Challenge for Mental Clarity Involving Radii, Perpendiculars, Midpoint

Challenging Geometry Puzzle: Problem 1592. 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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Geometry Problem 1590: Triangle, Incenter, Incircle, Tangency Points, Midpoints, Areas

Challenging Geometry Puzzle: Problem 1590. 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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Geometry Problem 1589: Prove the Ratio of BD to OD is Square Root of Two in a 90° Circular Sector

Challenging Geometry Puzzle: Problem 1589. 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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Garden Geometry 1585: The Hidden Harmony of Paths and Tangent Points

Challenging Geometry Puzzle: Problem 1585. 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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Geometric Problem1584: Proving a Ratio with the Incircle, Tangency Points, and Angle Bisector

Challenging Geometry Puzzle: Problem 1584. 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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Problem-Solving Challenge 1583: Prove Lines CK and BL are Parallel

Challenging Geometry Puzzle: Problem 1583. 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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Geometry Problem 1582: Prove That Angles AFD and AEF Are Equal in This Secant Problem.

Challenging Geometry Puzzle: Problem 1582. 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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Geometry Problem 1580: 60-Degree Triangle Challenge: Find AD with Excenter and Circumcircle Clues!

Challenging Geometry Puzzle: Problem 1580. 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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Geometry Problem 1578: Find the Area of a Bicentric Quadrilateral with Perpendicular Extensions of Opposite Sides

Challenging Geometry Puzzle: Problem 1578. 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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Geometry Problem 1575: Prove an Angle Bisector in a Triangle Involving an Altitude, Midpoint, and Excircle

Challenging Geometry Puzzle: Problem 1575. 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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