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