Question 136
The points $A_0, B_0, C_0$ are the feet of the altitudes in an acute-angled triangle $ABC$. The points $A_1, B_1, C_1$ are placed inside the triangle so that $\angle A_1BC = \angle A_1AB$, $\angle A_1CB = \angle A_1AC$, $\angle B_1CA = \angle B_1BC$, $\angle B_1AC = \angle B_1BA$, $\angle C_1BA = \angle C_1CB$, $\angle C_1AB = \angle C_1CA$. The points $A_2, B_2$, and $C_2$ are the midpoints of the segments $AA_1, BB_1$, and $CC_1$ respectively. Prove that the lines $A_0A_2, B_0B_2$, and $C_0C_2$ intersect at one point.