In $\triangle ABC$ with $AB=20,AC=16,BC=12$ (a right triangle), let $O,I$ be the circumcenter, incenter; a circle centered at $M$ is tangent to legs $AC,BC$ and to the circumcircle. What is the area of $\triangle MOI$?
The incircle has radius $4$, and $O$ (circumcenter) is the midpoint of hypotenuse $AB$: with $C=(0,0),A(16,0),B(0,12)$, $I=(4,4)$, $O=(8,6)$, $M=(t,t)$.
Tangency of $M$'s circle with the circumcircle: $OM=10-t$ gives $(t-8)^2+(t-6)^2=(10-t)^2$, so $t=8$, $M=(8,8)$.
Shifting $I$ to the origin: $O\to(4,2)$, $M\to(4,4)$; shoelace area $=\frac12|16-8|=\boxed4$.
