1. What is the Power of a Point?
The Power of a Point describes a set of relationships between the lengths of line segments drawn from a point to a circle.
These relationships stay true no matter where the point lies—inside, on, or outside the circle.
It connects tangents, secants, and chords in a single unified theory.
2. When the Point is Outside the Circle
If a point lies outside the circle, you can draw either:
- Two secants from the point
- One tangent and one secant
Both cases follow strong geometric rules.
2.1. Secant–Secant Theorem
If two secants drawn from point \(P\) meet the circle at \(A, B\) and \(C, D\), such that:
\(P-A-B\) and \(P-C-D\)
then the theorem states:
PA \times PB = PC \times PD
2.2. Tangent–Secant Theorem
If a tangent \(PT\) and a secant \(P-A-B\) are drawn from point \(P\), then:
PT^2 = PA \times PB
This is one of the most important formulas in circle geometry.
3. When the Point is Inside the Circle
If a point lies inside the circle, and two chords intersect at the point, the Power of a Point gives a product rule.
3.1. Chord–Chord Theorem
If chords \(AB\) and \(CD\) intersect at point \(P\) inside the circle, then:
\(P-A-B\)
\(P-C-D\)
The theorem says:
PA \times PB = PC \times PD
This product remains constant for that point.
4. Meaning of the Word 'Power'
Here, 'power' does not mean electricity or energy. In geometry, the power of a point is simply the constant value of these segment products for a given point.
For example, for point \(P\):
\text{Power of } P = PA \times PB
5. Using Power of a Point in Problems
This theorem is extremely useful for finding unknown lengths in circle diagrams involving tangents, secants, and intersecting chords.
5.1. Example 1: Tangent–Secant Case
If a tangent from point \(P\) has length \(PT = 9\text{ cm}\), and a secant intersects the circle at \(A\) and \(B\) such that \(PA = 6\text{ cm}\), then:
PT^2 = PA \times PB
9^2 = 6 \times PB
81 = 6PB
PB = 13.5\text{ cm}
5.2. Example 2: Secant–Secant Case
Two secants from \(P\) meet the circle at:
\(A = 4\text{ cm}, B = 10\text{ cm}\)
\(C = 3\text{ cm}, D = x\)
Using the theorem:
PA \times PB = PC \times PD
4 \times 10 = 3 \times x
40 = 3x
x = \dfrac{40}{3}\text{ cm}
6. Real-Life Connections
Power of a Point ideas appear in:
- Optics and ray tracing in circular lenses
- Designing circular arches and structures
- Navigation and range-finding from fixed points
- Robotics path calculations around circular zones