NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.3 – Part 2
Q3 If two circles intersect at two points; prove that their centres lie on the perpendicular bisector of the common chord.
Solution:

A and B are the two points at the intersection of two circles.

To prove, AB is bisector and OO’ is perpendicular to OO’.
In and ,
Thus, (by corresponding parts of congruent triangle)
In ΔAOC and ΔBOC,
Thus, (by corresponding parts of congruent triangles)
Also,
Hence, OO' is perpendicular to AB. Since , therefore AC = CB, i.e. C is the midpoint of AB. Therefore, OO' is perpendicular bisector of AB.