I would imagine a circle drawn on the inside would be 1/2 the side of the next circle. It's all related to the fact that the two radii are related to one another with the square root of two. That's based on the fact the triangle formed by the two radii is a 45/45/90 triangle, and the hypoteneus of a such a triangle is the square root of 2 times bigger than the adjacent leg.
In 10th grade I genuinely feared failing geometry class because I could not get proofs. Then one day it clicked. I was okay at them up to a point. But never a superstar.
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