jmc

$△DEF$ is inscribed inside $△ABC$ such that $D,E,F$ lie on $BC,AC,AB$, respectively. The circumcircles of $△DEC,△BFD,△AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB=23,BC=25,AC=24$, and $\stackrel{⌢}{\mathrm{BF}}=\stackrel{⌢}{\mathrm{EC}},\stackrel{⌢}{\mathrm{AF}}=\stackrel{⌢}{\mathrm{CD}},\stackrel{⌢}{\mathrm{AE}}=\stackrel{⌢}{\mathrm{BD}}$. The length of $BD$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.