The perpendicular dropped from the center of a circle on a chord bisects the chord.
Consider the equation of the circle
Let the points and be the ends of the chord as shown in the given diagram. Since the circle passes through the point , the equation of the circle becomes
Also the equation of the circle passes through the second point , so the circle becomes
Suppose that is the midpoint of the chord , then by using the midpoint formula we have
The slope of the chord is given by
The slope of the perpendicular
The equation of the line passing through the center and perpendicular to the chord is
It is observed that if 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 center of the circle bisects the chord.
Conversely, this can prove that the perpendicular bisector of any chord of a circle passes through the center of the circle.