DE-BUNDESWETTBEWERB 2017 Problem 3

Let $M$ be the incenter of the tangential quadrilateral $A_1A_2A_3A_4$ . Let line $g_1$ through $A_1$ be perpendicular to $A_1M$ ; define $g_2,g_3$ and $g_4$ similarly. The lines $g_1,g_2,g_3$ and $g_4$ define another quadrilateral $B_1B_2B_3B_4$ having $B_1$ be the intersection of $g_1$ and $g_2$ ; similarly $B_2,B_3$ and $B_4$ are intersections of $g_2$ and $g_3$ , $g_3$ and $g_4$ , resp. $g_4$ and $g_1$ . Prove that the diagonals of quadrilateral $B_1B_2B_3B_4$ intersect in point $M$ .