ABCD is a parallelogram whose diagonals AC and BD intersect at O.

Question:

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(ΔPOA) = ar(ΔQOC).

Solution:

In triangles POA and QOC, we have

∠AOP = ∠COQ

AO = OC

∠PAC = ∠QCA

So, by ASA congruence criterion, we have

ΔPOA ≅ ΔQOC

⇒ ar(ΔPOA) = ar(ΔQOC).

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