Question

Given a convex quadrilateral ABCD. Each of its sides is divided into K equal parts. Points on side AB are connected to corresponding points on CD, and points on BC are connected to points on DA, creating K² smaller quadrilaterals. From these, K quadrilaterals are chosen such that any two quadrilaterals are separated by at least one line connecting AB and CD, and one line connecting BC and DA. Prove that the sum of the areas of these quadrilaterals is S_ABCD/K. 
By A. Angans.