Power of a point

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Geometric description of the power of a circle

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)2r2.

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

AP \cdot PB = CP \cdot PD.

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

AP \cdot PB = CP \cdot PD.

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.

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