BMO Round 1, 2007

Geometry Difficulty 3.0 AMC 10/12 Prove it United Kingdom

In the convex quadrilateral $ABCD$ , points $M, N$ lie on the side $AB$ such that $AM = MN = NB$ , and points $P, Q$ lie on the side $CD$ such that $CP = PQ = QD$ . Prove that
$\text{Area of } AMCP = \text{Area of } MNPQ = \frac{1}{3}\,\text{Area of } ABCD.$

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