IMO 1988 LL SPA76

Origin: SPA

Problem

The quadrilateral A1A2A3A4 is cyclic and its sides are a1 = A1A2, a2 = A2A3, a3 = A3A4, and a4 = A4A1. The respective circles with centers Ii and radii \rhoi are tangent externally to each side ai and to the sides ai+1 and ai−1 extended (a0 = a4). Show that

i=1 ai \rhoi = 4(csc A1 + csc A2)2.