HMMT-Feb 2016 geo, Problem 4

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 $B A T$, MAN intersect at $D$. Compute $D C$.

Solution: We note that $D$ is the circumcenter $O$ of $A B C$, since $2 \angle C=\angle A T B=\angle A O B$. So we are merely looking for the circumradius of triangle $A B C$. By Heron's Formula, the area of the triangle is $\sqrt{9 \cdot 6 \cdot 1 \cdot 2}=$ $6 \sqrt{3}$, so using the formula $\frac{a b c}{4 R}=K$, we get an answer of $\frac{3 \cdot 8 \cdot 7}{4 \cdot 6 \sqrt{3}}=\frac{7 \sqrt{3}}{3}$. Alternatively, one can compute the circumradius using trigonometric methods or the fact that $\angle A=60^{\circ}$. $\fbox{\frac{7 \sqrt{3}}{3}}$.

Answer: \frac{7 \sqrt{3}}{3}