KöMaL 2009 B4149

The orthogonal projections of a point onto the lines of the altitudes drawn from vertices $A$, $B$, $C$ of a triangle are denoted by $A_1$, $B_1$, $C_1$, respectively. Prove that there is exactly one point in the plane of the triangle for which the line segments $AA_1$, $BB_1$, $CC_1$ are equal, and then the length of these line segments is equal to the diameter of the inscribed circle of triangle ABC .