The perpendicular dropped from the centre of a circle on a chord bisects the chord.
Consider the equation of the circle
Let the points and be ends of the chord as shown in the given diagram. Since the circle passes through the point so equation of circle becomes
Also equation of circle passes through the second point , so circle becomes
Suppose that be the midpoint of the chord , then by using midpoint formula we have
Slope of the chord is given by
Slope of perpendicular
Equation of line passing through the centre and perpendicular to the chord is
It is observed that whether the perpendicular (iv) bisects the chord, we check whether satisfies (iv). Putting the values in equation (iv), we get
Thus, satisfies equation (iv), so the perpendicular dropped from the centre of the circle bisects the chord.
Conversely, this can be proves that the perpendicular bisector of any chord of a circle passes through the centre of the circle.