**If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.**

**Solution:**

Draw two circles with centres O and O'. Join A and B to get a common chord AB. Join O and O' with the mid-point M of AB.

We need to prove that centres lie on the perpendicular bisector of the common chord. In other words, we need to prove that OO' is a straight line and .

In , M is the mid-point of chord AB.

. **(1)**

**(The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.)**

Similarly, in , M is the mid-point of chord AB.

**(2)**

**(The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.)**

From **(1)** and **(2)**, we can say that which means that OO' is a straight line and we can say that centres lie on the perpendicular bisector of the common chord.