power of a circle

The power of a fixed point A with respect to a circle of radius r and center O is defined by the product p congruent A P×A Q, where P and Q are the intersections of a line through A with the circle. The term "power" was first used in this way by Jacob Steiner. Amazingly, p (sometimes written k^2) is independent of the choice of the line A P Q. Now consider a point P not necessarily on the circumference of the circle. If d = O P is the distance between P and the circle's center O, then the power of the point P relative to the circle is p = d^2 - r^2.

chord | chordal theorem | circle function | coaxal circles | inverse points | inversion circle | inversion radius | inversive distance | radical line | triangle power