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Concept
Power of a point
Summary
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.
Besides the properties mentioned in the lead there are further properties:
For any point outside of the circle there are two tangent points on circle , which have equal distance to . Hence the circle with center through passes , too, and intersects orthogonal:
The circle with center and radius intersects circle orthogonal.
If the radius of the circle centered at is different from one gets the angle of intersection between the two circles applying the Law of cosines (see the diagram):
( and are normals to the circle tangents.)
If lies inside the blue circle, then and is always different from .
Official source
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