Tangent-Secant Theorems: Power of a Point Explained Simply

A tangent touches a circle at exactly one point, while a secant cuts through it at two points. When a tangent and a secant (or two secants) are drawn from the same outside point, their lengths follow beautifully simple multiplication rules.

These tangent-secant theorems let you find unknown lengths around a circle without angle chasing or trigonometry. They are collected under one elegant idea called the power of a point, and they appear in board exams, Olympiads, and engineering drawings alike.

Key Concepts

• Tangent: touches the circle at one point and is perpendicular to the radius there.
• Secant: meets the circle at two points; the part outside is the external segment, and the whole line from the external point to the far intersection is the whole secant.
• Power of a point: for a fixed external point P, the product (external segment × whole secant) is the same for every secant from P — and equals the square of the tangent length.
• Internal intersection: when two chords intersect inside the circle at X: AX × XB = CX × XD.
• Units: all lengths in the same linear units; the power of a point has area units since it is a product of two lengths.

Formulas

• Tangent-secant theorem: PT² = PA × PB, where from external point P the tangent touches at T and the secant meets the circle at A (near) and B (far). P must be outside the circle.
• Two secants from one external point: PA × PB = PC × PD (A and C nearer to P).
• Two intersecting chords inside: if chords AB and CD intersect at X inside the circle, then AX × XB = CX × XD.

Worked Example

From an external point P, a tangent to a circle measures 6 cm. A secant from P passes through the circle, and its external part (from P to the near intersection) is 4 cm. Find the length of the whole secant and the internal (chord) part.

1. Apply the tangent-secant theorem: PT² = external × whole, so 6² = 4 × whole.
2. 36 = 4 × whole, so whole = 36 / 4 = 9 cm.
3. The internal part (inside the circle) = whole − external = 9 − 4 = 5 cm.
4. Check: 4 × 9 = 36 = 6². Correct.

Common Misconceptions

• It is external × whole, not external × internal. The most common error is multiplying the outside piece by the inside piece; you need the entire secant from the external point to the far intersection.
• The tangent length is squared. Writing PT = PA × PB instead of PT² = PA × PB breaks both the units and the answer.
• The external point must be outside the circle. These product rules do not apply to a point inside the circle — there the intersecting-chords rule is used instead.

Real-World Applications

• Optics: tangent-secant geometry describes how light grazes curved mirrors and lenses, used in telescope design.
• Engineering: belt drives around pulleys use tangent lines, and the theorems help compute belt lengths around circular wheels.
• Navigation: the power-of-a-point idea underlies distance calculations from a known point to circular ranges, similar to GPS trilateration.

The power of a point unifies everything: from any external point, tangent² equals external segment times whole secant, and any two secants from that point give equal products. Inside the circle, intersecting chords split into segments with equal products. Remember to square the tangent and to use the whole secant, and these theorems solve circle-length problems in one step.