ISL 1980 Problem 8

Three points $A,B,C$ are such that $B \in ]AC[$ . On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$ . The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$ . Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$ . Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$ .