ABCD is a parallelogram whose diagonals AC and BD intersect at O. A line through O intersects AB at P and DC at Q. Prove that

ar(Δ POA) = ar(Δ QOC)

In ΔPOA and QOC, we have

AOP = COQ (Vertically opposite angle)


OA = OC (Diagonals of parallelogram bisect each other)


PAC = QCA (AB DC, alternate angles)


So, by ASA congruence rule, we have


ΔPOA ΔQOC


Area (ΔPOA) = Area (ΔQOC)


Hence, proved



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