Power of a Point

Every line through P meets the circle in the same product: PX · PY is an invariant.

The idea

Theorem (Power of a point).

Fix a circle and a point $P$. If a line through $P$ meets the circle at $X$ and $Y$, then the product $PX \cdot PY$ is the same for every such line.

This common value is the power of $P$ with respect to the circle: it depends on the point and the circle, not on the line.

The theorem covers three configurations at once. If $P$ lies inside the circle, two chords through it satisfy $PA \cdot PB = PC \cdot PD$. If $P$ lies outside, two secants from it satisfy the same equation, each product pairing the near and the far intersection along one line. A tangent from $P$ is the limiting secant whose two intersections have merged at the point of contact $T$, so its product is $PT \cdot PT$ and the tangent length satisfies $PT^{2} = PA \cdot PB$.

The figure shows the chord case, with the segments $AC$ and $DB$ drawn in; the walkthrough uses the two triangles they create to prove it.

Ways to work on it

• Walkthrough. Prove the chord case with inscribed angles and AA similarity.
• Practice. Compute chord, secant, and tangent lengths.
• Hardest. Find an unknown length from a tangent and a secant, then a circle's radius.

Not sure where to start? Take the ten-question placement test.