Question 148
Let a triangle $A_1A_2A_3$ and a point $X$ which does not lie on any of the lines $A_1A_2$, $A_1A_3$ and $A_2A_3$ be given. Denote by $l_i$ the line which passes through the point $A_i$ and is perpendicular to the line $XA_i$, $i=1,2,3$. Let $B_1 = l_2 \cap l_3$, $B_2 = l_3 \cap l_1$, $B_3 = l_1 \cap l_2$ be the intersection points, and let $H_1, H_2, H_3$ be the orthocenters of the triangles $B_1A_2A_3$, $B_2A_3A_1$, $B_3A_1A_2$, respectively. Prove that the triangle $H_1H_2H_3$ is congruent to the triangle $A_1A_2A_3$.
Fig. 10
Fig. 10