# Perpendicular from any Point of a Circle on a Diameter is a Mean Proportional

The perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which it divides the diameter.

Consider the equation of the circle with a center at the origin is given by the equation

Let be any point on the given circle. Let be the diameter of the circle as shown in the given diagram. From draw a perpendicular on diameter at . The coordinates of are because lies on the X-axis and lies on the X-axis. The coordinates of and are and respectively as shown in diagram.

Now we shall find the distance between and using the distance formula as follows:

Also

Now

From equation (i), we have . Putting this value in equation (ii), we have

This shows that the perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which the diameter is divided.