Question

If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ.

Answer 1

We draw two circle with centers `C_(1)` and `C_(2)` passing through the end points P and Q of a line segment PQ. We know, that perpendicular bisectors of a chord of a circle always passes through the centre of circel.
Thus, perpendicular bisector of PQ passes through `C_(1)` and `C_(2)`. Similarly, all the circle passing through PQ will have their centre on perpendiculars bisectors of PQ.