Question 15
Suppose the excircle of triangle $ABC$ opposite $A$ touches side $BC$ at $A_1$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly using the excircles opposite $B$ and $C$. Suppose the circumcenter of triangle $A_1B_1C_1$ lies on the circle through $A,B,C$. Prove that triangle $ABC$ is right-angled.