Definition
The Power of a Point refers to a set of theorems describing the proportional relationships between segments of chords, secants, and tangents that intersect at a specific point.
Why It Matters
These theorems prove that circles have “invisible rules” that govern intersections. If you are calculating the distance to the horizon from a plane or designing a lens, these invariant products () are your “Guardrails.” They ensure that geometry remains predictable even when lines and circles collide at arbitrary angles.
Core Concepts
- Chord-Chord Product Theorem (6.3.5): If two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other:
- How to read: “The length a times the length b is equal to the length c times the length d.”
- Meaning: Intersecting chords—product of segment lengths on one chord equals product on the other.
- Secant-Secant Theorem (6.3.6): If two secant segments are drawn from an external point, the product of the length of the entire secant and its external part is constant:
- How to read: “The length of the first whole secant segment times its external part is equal to the length of the second whole secant segment times its external part.”
- Meaning: From an external point, both secants have the same power-of-a-point product.
- Tangent-Secant Theorem (6.3.7): If a tangent and a secant are drawn from an external point, the square of the tangent length equals the product of the entire secant and its external part:
- How to read: “The length of the tangent segment squared is equal to the length of the whole secant segment times its external part.”
- Meaning: Tangent length squared equals secant’s full length times its external segment.