Solution: Comparing line segments in two circles.

Construct PC parallel to AB. Consider the right triangle OCP. Now the length of
OP = a + b and OC = a - b. So

From triangle OCP we know that with equality if and only if a = b.

Since CP = AB, we have a proof of the Arithmetic Mean -- Geometric Mean theorem for two positive values.  That is for positive a and b,

with equality if and only if a = b.