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 C with centre O and radius r is defined as

$(AO)^2 - r^2.\!$

Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero.

The power of a point theorem, due to Jakob Steiner, states that for any line through A intersecting C in points P and Q, the power of the point with respect to the circle is given up to a sign by the product

$AP \cdot AQ$

of the lengths of the segments from A to P and A to Q, with a positive sign if A is outside the circle and a negative sign otherwise: if A is on the circle, the product is zero. In the limiting case, when the line is tangent to the circle, P=Q, and the result is immediate from the Pythagorean theorem.

In the other two cases, when A is inside the circle, or A is outside the circle, the power of a point theorem has two corollaries.

The theorem of intersecting chords (or chord-chord power theorem) states that if A is a point in a circle and PQ and RS are chords of the circle intersecting at A, then

$AP \cdot AQ = AR \cdot AS.$

The common value of these products is the negative of the power of the point A with respect to the circle.

The theorem of intersecting secants (or secant-secant power theorem) states that if PQ and RS are chords of a circle which intersect at a point A outside the circle, then

$AP \cdot AQ = AR \cdot AS.$

In this case the common value is the same as the power of A with respect to the circle.

The power of a point is a special case of the Darboux product between two circles, which is given by

$(A_1A_2)^2-r_1^2-r_2^2$

where A₁ and A₂ are the centres of the two circles and r₁ and r₂ are their radii. The power of a point arises in the special case that one of the radii is zero. If the two circles intersect, then their Darboux product is

$r_1 r_2 \cos\varphi$

where φ is the angle of intersection.

[edit] References

Coxeter, H. S. M. (1969), Introduction to Geometry (2nd ed.), New York: Wiley .
Darboux, Gaston (1872), “Sur les relations entre les groupes de points, de cercles et de sphéres dans le plan et dans l’espace”, Annales Scientifique de l'École Normale Superieure 1: 323–392 .
Steiner, Jakob (1826), “Einige geometrische Betrachtungen”, Journal für die reine und angewandte Mathematik 1: 161–184 .