Mock AIME 1 2006-2007 Problems/Problem 10

10. In $\triangle ABC$ , $AB$ , $BC$ , and $CA$ have lengths $3$ , $4$ , and $5$ , respectively. Let the incircle, circle $I$ , of $\triangle ABC$ touch $AB$ , $BC$ , and $CA$ at $C'$ , $A'$ , and $B'$ , respectively. Construct three circles, $A''$ , $B''$ , and $C''$ , externally tangent to the other two and circles $A''$ , $B''$ , and $C''$ are internally tangent to the circle $I$ at $A'$ , $B'$ , and $C'$ , respectively. Let circles $A''$ , $B''$ , $C''$ , and $I$ have radii $a$ , $b$ , $c$ , and $r$ , respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$ .

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Mock AIME 1 2006-2007 Problems/Problem 10