Question

ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH.

Answer 1

ABCD is a parallelogram with E and F as the mid-points of AB and CD respectively.

We need to prove that

Since E and F are the mid-points of AB and CD respectively.

Therefore,

,

And

,

Also, ABCD is a parallelogram. Therefore, the opposite sides should be equal.

Thus,

Also, (Because )

Therefore, BEFC is a parallelogram

Then, and …… (i)

Now,

Thus, (Because as ABCD is a parallelogram)

We get,

AEFD is a parallelogram

Then, we get:

…… (ii)

But, E is the mid-point of AB.

Therefore,

Using (i) and (ii), we get:

Hence proved.