Power of a point

In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826.

Specifically, the power of a point with respect to a circle with center and radius is defined by

If is outside the circle, then ,
if is on the circle, then and
if is inside the circle, then .

Due to the Pythagorean theorem the number has the simple geometric meanings shown in the diagram: For a point outside the circle is the squared tangential distance of point to the circle .

Points with equal power, isolines of , are circles concentric to circle .

Steiner used the power of a point for proofs of several statements on circles, for example:

  • Determination of a circle, that intersects four circles by the same angle.
  • Solving the Problem of Apollonius
  • Construction of the Malfatti circles: For a given triangle determine three circles, which touch each other and two sides of the triangle each.
  • Spherical version of Malfatti's problem: The triangle is a spherical one.

Essential tools for investigations on circles are the radical axis of two circles and the radical center of three circles.

The power diagram of a set of circles divides the plane into regions within which the circle minimizing the power is constant.

More generally, French mathematician Edmond Laguerre defined the power of a point with respect to any algebraic curve in a similar way.

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