Question

Abcd is parallelogram ,e and f are mid point of ab and cd . Gh is any line intersecting ad ,ef and bc at g, p and h. Prove gp=ph

Answer 1

Given: ABCD is a parallelogram, E and F are mid points of AB and DC and a line GH intersects AD, EF and BC at G, P, and H respectively.

Construction: Join GB and Let it intersect EF of X.

Now since E and F are mid points of AB and DC

⇒ AE = EB = AB
and DF= FC = DC
but AB = DC
⇒ AEFD and BCFE are parallelogram
⇒ AD EF BC ........ (1)

Now In ∆ ABG
AE = EB and EX AG (from (1))
⇒ X is the mid point of GB (mid point theorem)

and in ∆ GBH
GX = XB (∵ X is the mid point of GB)
and XP BH (from (1))
⇒ P is the mid point of GH
⇒ GP = PH