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Japanese Temple Problem

You are given two circles: \(O_1\) with radius \(r_1\) and \(O_2\) with radius \(r_2\), where \(r_1 > r_2\). They are tangent to each other and mutually tangent to a common line.  If \(O_1\) is tangent to the line at point \(A\) and \(O_2\) tangent at point \(B\), show that \((AB)^2 = 4r_1r_2\).

Sangaku problem, Miyagai Prefecture, 1892

"Japanese Temple Problem," Convergence (March 2006)