Power of a point
From Wikipedia, the free encyclopedia
The power of a point A (circle power, power of a circle) with respect to a circle with center 0 and radius r is defined as
- (AO)2 − r2.
Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero.
The theorem of intersecting chords (or chord-chord power theorem) states that if P is a point in a circle and AB and CD are chords of the circle intersecting at P, then
The common value of these products is the negative of the power of the point P with respect to the circle (since the power is negative and the product of positive lengths is positive).
The theorem of intersecting secants (or secant-secant power theorem) states that if AB and CD are chords of a circle which intersect at a point P outside the circle, then
In this case the common value is the same as the power of P because both are positive.
When expressed in this form it becomes clear that both theorems are really special cases of the more general power of a point theorem, which covers both these cases as well as the limiting cases where two points coincide and their secant becomes a tangent.
[edit] External links
- Power of a Point Theorem at cut-the-knot
- Eric W. Weisstein. "Circle Power." From MathWorld--A Wolfram Web Resource
- Intersecting Chords Theorem at cut-the-knot
- Intersecting Chords Theorem With interactive animation
- Intersecting Secants Theorem With interactive animation
