Question 132
Let $H$ be the point of intersection of the altitudes of the acute-angled triangle $ABC$. The points $A_1, B_1, C_1$ are the midpoints of the sides $BC, CA$, and $AB$ respectively. Let $A_2$ and $C_2$ be the points for which $A_2A \perp AC$ and $A_2C_1 \perp AB$, $C_2C \perp AC$ and $C_2A_1 \perp BC$. Prove the following:
a) the midpoint of the segment $BH$ lies on the line $A_2C_2$;
b) let the line $BB_1$ intersect the circle circumscribed about the triangle $A_1B_1C_1$ at the points $B_1$ and $B_3$; then the point $B_3$ lies on the line $A_2C_2$.
a) the midpoint of the segment $BH$ lies on the line $A_2C_2$;
b) let the line $BB_1$ intersect the circle circumscribed about the triangle $A_1B_1C_1$ at the points $B_1$ and $B_3$; then the point $B_3$ lies on the line $A_2C_2$.