Question 76
On the plane, let $ABC$ be the acute and non-isosceles triangle. Let $B', C'$ be the symmetric points of $B, C$ with respect to the lines $CA, AB$, respectively. Assume that $BC', B'C$ meet at $A_0$. Let $A_1$ be the circumcenter of triangle $AB'C'$ and $A_2$ be the center of the circle passing through the projections of $A_0$ on the lines $BC, CA, AB$. The points $B_1, B_2, C_1, C_2$ are defined similarly.
Prove that $A_1A_2, B_1B_2, C_1C_2$ are concurrent at the Euler point of the triangle $ABC$.
Prove that $A_1A_2, B_1B_2, C_1C_2$ are concurrent at the Euler point of the triangle $ABC$.