IZhO 2014 Problem 6

Let $A B C D$ be a convex quadrilateral, partitioned by four lines as in the picture below, such that the meeting points of these lines all lie on the diagonals of $A B C D$. Prove that if the corner quadrilaterals 1, 2,3 and the center quadrilateral 4 are all tangential, then the corner quadrilateral 5 is also tangential.