Theorem: (Pythagorean Power Formula)
      Let P be any point and c be any circle with radius r. Let l be any line through the center of c and E be a point on line l. Then P(c) = E(c) + h2, where h = PE, iff PE is perpendicular to l.
    Proof:
      By definition of the power of a point with respect to circle c with center O, P(c) = PO2 - r2 and E(c) = EO2 - r2.

      When the two equations are subtracted, r2 cancels out. The result is P(c) - E(c) = PO2 - EO2.

      By the Pythagorean Theorem, PO2 = EO2 + h2 . Therefore, h2 = PO2 - EO2.

      Substitute PO2 -EO2 for h2 into the equation P(c) - E(c) = PO2 - EO2.

      Therefore, P(c) = E(c) + h2.

                        QED