Question

Prove that the perpendicular bisector of a chord of a circle passes through the center.

Answer 1

Let us support PQ does not pass through O.
Since $PQ\perp$ bisector of $AB$
$\angle QOP=90°⟶\left(1\right)$
$P$ is the mid-point of $AB.$
$QP$ passes through center from this, we can say
$OP\perp AB\angle OPA=90°.........\left(2\right)$
Comparing 1 $\&$ 2, we can say
$\therefore \angle QOP=\angle OPA,$

This can be possible only when $O,P$ and $Q$ are on the same line.
$\therefore$ It is a contradiction $PQ$ passes through $O.$

Thus , $PQ$ must passes through $O$