Tangent–secant length theorem

From an external point, tangent² = external × whole secant.

Key formulas

Tangent touches at T. Secant from P cuts the circle at A (first) and B (farther).

PT=6, PA=4 → 36=4×PB → PB=9. Then the internal chord AB=5.

If both lines are tangents, the lengths from P are equal — a different but related fact.

Problem. From point P, a tangent touches a circle at T with PT = 6. A secant from P cuts the circle at A then B with PA = 4. Find PB.

Solution. PT² = PA × PB → 36 = 4 × PB → PB = 9. The full secant length is 9.

This is one case of the power of a point theorem.

PT=8, PA=5. Then PB is:

A is the nearer intersection; PB is the full secant length from P.