Let M be the incenter of the tangential quadrilateral A1A2A3A4. Let line g1 through A1 be perpendicular to A1M; define g2,g3 and g4 similarly. The lines g1,g2,g3 and g4 define another quadrilateral B1B2B3B4 having B1 be the intersection of g1 and g2; similarly B2,B3 and B4 are intersections of g2 and g3, g3 and g4, resp. g4 and g1.
Prove that the diagonals of quadrilateral B1B2B3B4 intersect in point M.