Question

In a parallelogram ABCD, E and F are the midpoints of the sides AB and CD respectively. The line segments AF and BF meet the line segments DE and CE at points G and H respectively Prove that: EGFH is a parallelogram.

Answer 1

In quadrilateral AECF,
AE = CF   ....[from (i)]
AE || CF   ....[as AB || DC]
⇒ AECF is a parallelogram.
⇒ EC || AF or EH || GF      ....(i)
In quadrilateral BFDE,
BE = DF  ....[from (i)]
BE || DF  ....[as AB || DC]
⇒ BEDF is a parallelogram  
⇒ BF || ED or HF || EG     ....(ii)
From (i) and (ii), we get that
EGFH is parallelogram.