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The infinite snowman


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Question

Consider an isosceles triangle with base 10 feet and sides 13 feet. Now imagine building a snowman inside that triangle. A snowman made of circles, not spheres. It's easy to sketch. The bottom circle is tangent to all three sides. The next circle rests on the first and is tangent to the two sides. Likewise the third, fourth, and so on. Actually, there are an infinite number of circles in our snowman. But there must be a geometric series involved, or something like one, because the snowman never gets outside the triangle. So, it's fair to ask:

What is the sum of the circumferences of all the circles?

This is actually an unadvertised "aha" puzzle, so can you give the answer without writing anything down except for the sketch?

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