Question

$E$ and$F$ are points on diagonal $AC$ of a parallelogram $ABCD$ such that $AE=CF$. Show that$BFDE$ is a parallelogram.

Answer 1

Step 1: Construction.

Join $BD$, meeting $AC$ at $O$.

$E$ and$F$ are points on diagonal $AC$ of a parallelogram $ABCD$ such that $AE=CF$

Step 2: Prove that$BFDE$ is a parallelogram.

Since diagonals of a parallelogram bisect each other,

We note that $OA=OC\&OD=OB.$

And, $OA=OCandAE=CF,$

$\Rightarrow$$OA–AE=OC–CF$

$\Rightarrow$ $OE=OF$

So,$BFDE$ is a quadrilateral whose diagonals bisect each other.

Hence,$BFDE$ is a parallelogram.