Question 74
Let $\triangle ABC$ be acute triangle and let $A_1, B_1, C_1$ be points on its sides $\overline{BC}, \overline{CA}, \overline{AB}$ respectively. Prove that the triangles $\triangle ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1$, $\angle B = \angle B_1$, $\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ coincides with the circumcentre of the triangle $\triangle ABC$.