I wanted to start the year off with this, until I realized (duh) that it really uses the Pythagorean Theorem.

But it’s a great problem, and so rich for spinning off. What about triangles? What about pentagons? What about irregular triangles? Can you always gets a circle around the polygon, no matter where you start? (An honest-to-god context for drawing a circle through three points.)

More than ever, I’d like to really focus on problem posing this year, and this problem is very rich for jumping off from. I love it, and can’t wait to bring it in to class.

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This problem can be solved without knowing Pythagoras’s theorem.

SO SPEAKETH FIVE TRIANGLES

Hint: If you rotate every other square by 45 degrees, it becomes apparent that the ratio of areas of successive squares is 2:1. If there were another square drawn around the largest circle, the ratio of areas of the largest square to the smallest square would be 16:1, and their sides would be in the ratio 4:1.

But the side of each square is the diameter of the circle it contains. So the length of the radius of the largest circle is 4 times the length of the radius of the smallest circle, and their areas would be in ratio 16:1.