SHARYGIN 2013 Problem 3

Let $ABC$ be a right-angled triangle $(\angle B = 90^\circ)$. The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1$, $A_2$ respectively; points $C_1$, $C_2$ are defined similarly. Prove that the perpendiculars from $A$, $B$, $C$ to $C_1C_2$, $A_1C_1$, $A_1A_2$ respectively, concur.