Let $ABCD$ be a convex quadrilateral, and the points $A_1 \in (CD)$, $A_2 \in (BC)$, $C_1 \in (AB)$, $C_2 \in (AD)$. Let $M, N$ be the intersection points between the lines $AA_2, CC_1$ and $AA_1, CC_2$ respectively. Prove that if three of the quadrilaterals $ABCD$, $A_2BC_1M$, $AMCN$, $A_1NC_2D$ are circumscriptive (i.e. there exists an incircle tangent to all the sides of the quadrilateral) then the forth quadrilateral is also circumscriptive.