Question 55
Let $k$ be a circle with diameter $AB$ and let $C \in k$ be an arbitrary point. The excircles of $\triangle ABC$ tangent to the sides $AC$ and $BC$ are tangent to the line $AB$ at points $M$ and $N$, respectively. Denote by $O_{1}$ and $O_{2}$ the circumcenters of $\triangle AMC$ and $\triangle BNC$. Prove that the area of $\triangle O_{1}CO_{2}$ does not depend on $C$.