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The total length $L$ is $4\sqrt{Rr} + \pi(R + r) + 2(R -r)\sin^{-1}(R - r)/(R + r).$
Belt
Age 16 to 18
Challenge Level 
Herbert of Sha Tin College, Hong Kong submitted the only correct solution to date which is given below. Can anyone give an alternative solution?
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$\alpha = \sin^{-1}(R - r)/(R + r)$
$L_1 = R(\pi + 2\alpha)$
$L_4 = r(\pi - 2\alpha)$
$L_2 = L_3 = x$
$x^2 = (R + r)^2 - (R - r)^2$
$x^2 = 4Rr$
$x = 2\sqrt{Rr}$
$L_2 + L_3 = 4\sqrt{Rr}$
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The total length $L$ is $4\sqrt{Rr} + \pi(R + r) + 2(R -r)\sin^{-1}(R - r)/(R + r).$
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