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

$\huge\underline{\overline{\mid{\bold{\pink{Answer:-}}\mid}}}$

ANSWER

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

∴AE=BE=21AB and CF=DF=21CD

But, AB=CD

∴21AB=21CD⇒BE=CF

Also, BE∥CF     [∵AB∥CD]

∴ BEFC is a parallelogram.

⇒BC∥EF and BE=PH     ...(i)

Now, BC∥EF

⇒AD∥EF     [∵BC∥AD as ABCD is a ∥gm]

⇒AEFD is a parallelogram

⇒AE=GP     ...(ii)

But, E is the mid-point of AB.

∴AE=BE

⇒GP=PH     [Using (i) and (ii)]