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 (Δ POA) = ar (Δ QOC).

Answer 1

In triangles and , POA QOC we have

∠AOP  = ∠COQ             [vertically opposite angles]

       OA = OC                 [ Diagonals of a parallelogram bisect each other]

   ∠PAC = ∠QCA             [ AB || DC ; alternative angles ]

So, by ASA congruence criterion, we have

ΔPOA  ≅ QOC

Area (ΔPOA)  = area (ΔQOC) .