Let P and Q be the centers of two circle C and C, each passing through two given points A and B.
Then,
PA = PB (radii of the circle C)
⇒ P lies on the perpendicular bisector of AB ….(i)
Again QA = QB (radii of the circle C)
⇒ Q lies on the perpendicular bisector of AB … (ii)
From (i) and (ii) it follows that P and Q both. lie on the perpendicular bisector of AB.
Hence, the locus of the centers of all the circles passing through A and B is the perpendicular bisector of AB.