Question 13
Triangle $AB_0C_0$ has side lengths $AB_0 = 12,$ $B_0C_0 = 17,$ and $C_0A = 25.$ For each positive integer $n,$ points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}},$ respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1}$ $\sim \triangle AB_{n-1}C_{n-1}.$ The area of the union of all triangles $B_{n-1}C_nB_n$ for $n \ge 1$ can be expressed as $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $q.$