Question

In the figure, $ABCD$ is a parallelogram whose diagonals intersect each other at $O$. A line segment $EOF$ is drawn to meet $AB$ at $E$ and $DC$ at $F$. Prove that $OE=OF$.

Answer 1

We know that $ABCD$ is a parallelogram whose diagonals intersect each other at $O$

Consider $△AOE$ and $△COF$

• We know that $\angle CAE$ and $\angle DCA$ are alternate interior angles

$⟹\angle CAE=\angle DCA$

• From the figure we know that the diagonals are equal and bisect each other

$⟹AO=CO$

• We know that $\angle AOE$ and $\angle COF$ are vertically opposite angles

$⟹\angle AOE=\angle COF$

By ASA congruence criterion

$△AOE\cong COF$

$⟹OE=OF$ (By C.P.C.T)