Power of a Point

One number controls every line through P.

For a point $P$ and a circle of centre $O$, radius $r$, the power of $P$ is $OP^2 - r^2$. Every line through $P$ meeting the circle at $A$ and $B$ gives the same product $PA \cdot PB$, and the choice of line is irrelevant, and that invariance is the technique.

It wears three faces: two secants ($PA \cdot PB = PC \cdot PD$), a tangent and a secant ($PT^2 = PA \cdot PB$), and two chords crossing inside. All three are the same statement.

The locus of equal power to two circles is a line, the radical axis, which is why power arguments so often deliver collinearity or concurrency without any computation.

More techniques

• InvariantsFind what the chaos cannot touch.
• Symmetry Breaking & WLOGOrder the variables before you start.
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