$\triangle AEB: EB^{2} + (2r_{1})^{2} = AE^{2}$, see picture. $\triangle EBC: EB^{2} + (2r_{2})^{2} = EC^{2}$. $\Rightarrow 2 EB^{2} + 4r_{1}^{2} + 4r_{2}^{2}= AE^{2} + EC^{2} = AC^{2} (\triangle AEC) = (2 r_{1} + 2 r_{2})^{2}$ $EB = 2 \sqrt{r_{1}r_{2}}$. Then $ AE^{2} = 4r_{1}r_{2} + 4r_{1}^{2} \Rightarrow AE = 2\sqrt{r_{1}(r_{1}+r_{2})}$. and $ EC^{2} = 4r_{1}r_{2} + 4r_{2}^{2} \Rightarrow EC = 2\sqrt{r_{2}(r_{1}+r_{2})}$. Area $ \triangle AEC = 2 ( r_{1} + r_{2}) \cdot \sqrt{r_{1}r_{2}}$ $\triangle AUB \sim \triangle AEC: \frac{UB}{EC}=\frac{2r_{1}}{2r_{1}+2r_{2}} \Rightarrow EV = UB = 2\sqrt{r_{2}(r_{1}+r_{2})} \cdot \frac{r_{1}}{r_{1}+r_{2}}$ $\triangle BVC \sim \triangle AEC: \frac{BV}{AE}=\frac{2r_{2}}{2r_{1}+2r_{2}} \Rightarrow UE = BV = 2\sqrt{r_{1}(r_{1}+r_{2})} \cdot \frac{r_{2}}{r_{1}+r_{2}}$ Area $ \triangle EUV = \frac{EV \cdot UE}{2}= \frac{2 r_{1}r_{2}\sqrt{r_{1}r_{2}}}{r_{1}+r_{2}}$ $ R=\frac{[EUV]}{[EAC]} = \frac{r_{1}r_{2}}{(r_{1}+r_{2})^2} $

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