2018 AMC 12B 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 $\triangle MOI$ ?

This problem is part of the 2018 AMC 12B .