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Herbert of Sha Tin College, Hong Kong submitted the only correct solution to date which is given below. Can anyone give an alternative solution?
$\alpha = \sin^{-1}(R - r)/(R + r)$
$L_1 = R(\pi + 2\alpha)$
$L_4 = r(\pi - 2\alpha)$
$L_2 = L_3 = x$
$x^2 = (R + r)^2 - (R - r)^2$
$x^2 = 4Rr$
$x = 2\sqrt{Rr}$
$L_2 + L_3 = 4\sqrt{Rr}$
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