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Hint to puzzle 108: Eyeball to eyeball

Two circles, with centers O and P. From the center of each circle, two tangents are drawn to the circumference of the other. The tangents from O intersects that circle at A and B; the tangents from P intersects that circle at C and D. The tangent through A meets the other circle at S; the tangent through C meets the other circle at R. Line segment OP meets AB at E, and CD at F.

Use symmetry, and consider similar triangles.