GeoProof — Power of a Point explorer

PA · PB = PC · PD

Drag A, B, C, or D around the circle. P is where the two lines meet — inside (crossing chords) or outside (secants).

The Power of a Point

Fix a circle and a point P. Draw any line through P that meets the circle at two points — say A and B. The product of the two distances PA · PB is always the same, no matter which line you pick. So for a second line meeting the circle at C and D:

PA · PB = PC · PD

That shared value is called the power of the point P with respect to the circle.

Two faces: chords and secants

When P is inside the circle, the two lines are chords that cross at P — this is the "intersecting chords" theorem. When P is outside, the lines are secants and P is their external meeting point — the "secant–secant" theorem. The same equation covers both.

The exact value

If O is the center and r the radius, the power equals |PO² − r²|. Inside the circle PO < r, so the signed power PO² − r² is negative; outside it is positive. Either way the unsigned products PA · PB and PC · PD match it — toggle "Power value" to see PO² − r² alongside the products.

The tangent case

If P is outside and one line just grazes the circle, its two intersection points merge into a single tangent point T, and PA · PB becomes PT². So the tangent length satisfies PT² = PC · PD — a handy way to find tangent lengths.

A bit of history

The intersecting-chords and secant relations appear in Euclid's Elements (Book III, Propositions 35 and 36, c. 300 BCE). The unifying notion of the "power of a point" was named by Jakob Steiner in the 19th century.