AMC 12 2018b Problem 21

In $\triangle{ABC}$ with side lengths $AB = 13$, $AC = 12$, and $BC = 5$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$?

\textbf{(A)}\ \frac52 \textbf{(B)}\ \frac{11}{4} \textbf{(C)}\ 3 \textbf{(D)}\ \frac{13}{4} \textbf{(E)}\ \frac72