Question 86
Let $I, O, H, \Omega$ be the incenter, circumcenter, orthocenter and circumcircle of triangle $ABC$, respectively. Let $AI$ meet $\Omega$ at $M \neq A$, let $IH$ meet $BC$ at $D$, and let $MD$ meet $\Omega$ at $E \neq M$.
Prove that line $OI$ is tangent to the circumcircle of $\triangle IHE$.
Prove that line $OI$ is tangent to the circumcircle of $\triangle IHE$.