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 $PA \cdot PB$ — 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.

Where it appears on Lemma: Level 5 (circles & power of a point) and Level 8 configurations.

Train it on Lemma

Power of a Point unlocks at Level 5 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

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