Question

Prove that quadrilateral formed by the intersection of angle bisectors of all angle of parallelogram is a rectangle.

Answer 1

$ABCD$ is a parallelogram

The adjacent angles of a parallelogram sum up to give ${180}^{\circ }$

$i.e.,\angle B+\angle C={180}^{\circ }\frac{\angle B}{2}+\frac{\angle C}{2}=90$

From $\Delta OBC⟹\angle O={90}^{\circ }⟹\angle POR={90}^{\circ }$

Similarly, all the $4$ vetrices have right angles at their corners

Hence $PQRS$ is rectangle.