Power of A Point - Orthogonal Circle

Orthogonal Circle

For a point P outside the circle, the power h equals R2, the square of the radius R of a new circle centered on P that intersects the given circle at right angles, i.e., orthogonally (Figure 2). If the two circles meet at right angles at a point T, then radii drawn to T from P and from O, the center of the given circle, likewise meet at right angles (blue line segments in Figure 2). Therefore, the radius line segment of each circle is tangent to the other circle. These line segments form a right triangle with the line segment connecting O and P. Therefore, by the Pythagorean theorem,

where s is again the distance from the point P to the center O of the given circle (solid black in Figure 2).

This construction of an orthogonal circle is useful in understanding the radical axis of two circles, and the radical center of three circles. The point T can be constructed—and, thereby, the radius R and the power p found geometrically—by finding the intersection of the given circle with a semicircle (red in Figure 2) centered on the midpoint of O and P and passing through both points. By simple geometry, it can also be shown that the point Q is the inverse of P with respect to the given circle.