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).