Question 54
A straight line that contains the center of rectilinear triangle $ABC$ intersects straight lines $AB$, $BC$ and $CA$ in points $C_1$, $A_1$ and $B_1$ respectively. Let $A_2$ be the symmetric point to $A_1$ about the middle of $BC$; points $B_2$ and $C_2$ are defined in a similar manner. Prove that points $A_2$, $B_2$ and $C_2$ are on a line that touches the circle inscribed in the triangle $ABC$.
(Serdiuk Nazar)
Fig. 25
(Serdiuk Nazar)
Fig. 25