Power of A Point - Theorems

Theorems

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

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

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

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

• The tangent-secant theorem is a special case of the theorem of intersecting secants, where points Q and P coincide, i.e.

This has utility in such applications as determining the distance to a point P on the horizon, by selecting points R and S to form a diameter chord, so that RS is the diameter of the planet, AR is the height above the planet, and AP is the distance to the horizon.