Power of a Point
The power of a point \(P\) with respect to a circle centered at \(O\) is a measure of distance from the point to the circle, defined by
\[h = OP^2 - r^2,\]
which implies that
- points inside the circle have negative power;
- points on the circle have zero power;
- points outside the circle have positive power.
For points outside the circle, the power is equivalent to the square of the length of its tangent to the circle. It is also the square of the radius of an orthogonal circle centered at \(P\). Unfortunately, points inside the circle have no particularly good geometric interpretation.
The power of a point is used extensively in computational geometry, including in defining the radical axis (and thus radical center). It is better known for the power of a point theorem, which writes the power of a point in two different ways to show the equality between them. There are two cases to this theorem; the two secants form and the tangent-secant form.
Two secants:
Let \(\omega\) be a circle and \( P\) a point. Let \(m\) a line through \(P\) which meets \(\omega\) at point \(A\) and \(B\), and let \(n\) be another line through \(P\) which meets \(\omega\) at points \(C\) and \(D\). Then
\[PA\cdot PB=PC\cdot PD.\]
Tangent-secant:
There is a special case when \(P\) lies outside of the circle and one of the lines is a tangent, in which case
\[PA\cdot PB=PC^2.\]
Two circles \(O\) and \(Q\) are orthogonal: their radii are perpendicular at their intersection point \(P,\) as shown above. Then the line \(AD\) is drawn such that it passes through both centers and intersects the arcs at points \(B\) and \(C.\)
If \(AB = 10\) and \(CD = 24\), what is the length of the red segment \(BC?\)
