My 10th problem

Geometry Level 5

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 \(n\geq1\) can be expressed as \(\tfrac pq\), where \(p\) and \(q\) are relatively prime positive integers. Find \(q\).

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