KöMaL 2025 A912

Let $ABC$ be a triangle, and let the symmedian through $A$ intersect the circumcircle of the triangle at a second point $A'$. Prove that the following three points lie on a line:

• the reflection of point $A$ over the line $BC$;
• the inverse image of point $A$ under inversion with respect to the Feuerbach circle of triangle $ABC$;
• the midpoint of the chord $AA'$.