Question

In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P.
Prove that:
(i) BP = 2AD
(ii) O is the mid-point of AP.

Answer 1

Given ABCD is parallelogram, so AD = BC, AB = CD.

Consider triangle APB, given EC, is parallel to AP and E is the midpoint of side AB.
So by midpoint theorem,
C has to be the midpoint of BP.

So BP = 2BC, but BC = AD as ABCD is a parallelogram.
Hence BP = 2AD

Consider triangle APB, AB || OC as ABCD is a parallelogram.
So by midpoint theorem,
O has to be the midpoint of AP.
Hence Proved.