Question 84
Let $ABC$ be a triangle, let $A_1$, $B_1$, $C_1$ be the antipodes of the vertices $A$, $B$, $C$, respectively, in the circle $ABC$, and let $X$ be a point in the plane $ABC$, collinear with no two vertices of the triangle $ABC$. The line through $B$, perpendicular to the line $XB$, and the line through $C$, perpendicular to the line $XC$, meet at $A_2$; the points $B_2$ and $C_2$ are defined similarly. Show that the lines $A_1A_2$, $B_1B_2$ and $C_1C_2$ are concurrent.