Power of a point
From Freepedia
The power of a point P with respect to a circle with center C and radius r is defined as
- <math>(PC)^2 - r^2.</math>
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 inside a circle and AB and CD are chords of the circle intersecting at P, then
- <math>AP \cdot PB = CP \cdot PD.</math>
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
- <math>AP \cdot PB = CP \cdot PD.</math>
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.
