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

Hence it is proved that GP=PH.

Given,

ABCD is a parallelogram.

E and F are the mid-points of AB and CD

GH is any line intersecting AD,EF and BC at G, P and H

To prove: GP=PH

Consider the figure while going through the following steps.

Proof:

AE = BE = 1/2 AB

CF = DF = 1/2 CD

we have, AB = CD

⇒ 1/2 AB = 1/2 CD

⇒ BE = CF

and BE ║ CF

∴ BEFC is a parallelogram.

⇒ BC ║ EF

and BF = PH

Now, BC ║ EF

⇒ AD ║ EF

∴ AEFD is a parallelogram

⇒ AE = GP

E is the mid-point of AB

∴ AE = BE

⇒ GP = PH