Question 19
The excircle opposite $A$ in triangle $ABC$ is tangent to side $BC$ at $A_1$. Define $B_1$ on $CA$ and $C_1$ on $AB$ analogously. Suppose the circumcenter of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that $ABC$ is right-angled.
The excircle opposite $A$ is the circle tangent to side $BC$ and to the lines $AB,AC$, but not to the segments $AB,AC$. The excircles opposite $B,C$ are defined analogously.
The excircle opposite $A$ is the circle tangent to side $BC$ and to the lines $AB,AC$, but not to the segments $AB,AC$. The excircles opposite $B,C$ are defined analogously.