HMMT 2016 G4

Let $A B C$ be a triangle with $A B=3, A C=8, B C=7$ and let $M$ and $N$ be the midpoints of $\overline{A B}$ and $\overline{A C}$, respectively. Point $T$ is selected on side $B C$ so that $A T=T C$. The circumcircles of triangles $B A T, M A N$ intersect at $D$. Compute $D C$.