Question

Prove that the locus of centres of circles passing through points $A$ and $B$ is perpendicular bisector of line segment $AB$.

Answer 1

Let center of circel passing through two point $AB$ be $O$

Then $OA=OB$ because they are radius of the same circle

Hence $OAB$ is isosceles and altitude from $O$ to point $P$ on $AB$ define two congruent triangles $OPA$ and $OPB$

So $OP$ is perpendicular bisector of $AB$

Any point $O$ or centre of circle passing through two points is the perpendicular bisector of two points.