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Power of a Point (Circle) — Definition, Formula & Examples

The power of a point is a number associated with a point relative to a circle, defined by the product of signed distances along any line through that point that intersects the circle. Remarkably, this product is the same no matter which line you draw through the point.

Given a circle with center OO and radius rr, the power of a point PP with respect to the circle is defined as d2−r2d^2 - r^2, where d=∣OP∣d = |OP|. For any line through PP that intersects the circle at points AA and BB, the signed product PA⋅PBPA \cdot PB equals d2−r2d^2 - r^2, independent of the choice of line.

Key Formula

pow(P)=d2−r2=PA⋅PB\text{pow}(P) = d^2 - r^2 = PA \cdot PB
Where:
  • dd = Distance from point P to the center of the circle
  • rr = Radius of the circle
  • PA,PBPA, PB = Distances from P to the two points where any line through P meets the circle

How It Works

The theorem takes three common forms depending on where the point PP lies. If two chords intersect inside the circle at PP, then PA⋅PB=PC⋅PDPA \cdot PB = PC \cdot PD for the two chords ABAB and CDCD. If PP is outside the circle and two secants pass through it, the same product relationship holds using the distances to each pair of intersection points. If one of those secants becomes a tangent touching the circle at TT, the relationship simplifies to PT2=PA⋅PBPT^2 = PA \cdot PB. The power is negative when PP is inside the circle, zero when PP is on the circle, and positive when PP is outside.

Worked Example

Problem: Two chords intersect inside a circle at point P. One chord is divided by P into segments of length 3 and 8. The other chord is divided into segments of length 4 and x. Find x.
Apply the intersecting chords form: When two chords intersect inside a circle, the products of their segments are equal.
PA⋅PB=PC⋅PDPA \cdot PB = PC \cdot PD
Substitute known values: Plug in the given segment lengths.
3×8=4×x3 \times 8 = 4 \times x
Solve for x: Divide both sides by 4.
x=244=6x = \frac{24}{4} = 6
Answer: The unknown segment length is x=6x = 6.

Why It Matters

Power of a Point is a staple of competition math (AMC, MATHCOUNTS, olympiads) and appears frequently in SAT/ACT geometry problems involving intersecting chords and secants. It also forms the theoretical basis for the radical axis of two circles, a key tool in advanced Euclidean geometry.

Common Mistakes

Mistake: Using the wrong segments in the product — for example, multiplying the two segments of the same chord instead of the two segments on opposite sides of P along each chord.
Correction: Each product pairs the two distances from P to the circle along the same line. For a chord split by P into lengths aa and bb, the product is a⋅ba \cdot b, and this equals the corresponding product on the other chord.