Problem

Source: Sharygin 2005 finals 10.1

Tags: geometry, parallelogram, interior, convex quadrilateral



A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.