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.
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