Mock AIME 3 Pre 2005 Problems/Problem 11

Problem

$ABC$ is an acute triangle with perimeter $60$ . $D$ is a point on $\overline{BC}$ . The circumcircles of triangles $ABD$ and $ADC$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively such that $DE = 8$ and $DF = 7$ . If $\angle{EBC} \cong \angle{BCF}$ , then the value of $\frac{AE}{AF}$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$ .

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Mock AIME 3 Pre 2005 Problems/Problem 11

Problem

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