The Two-Tangent Theorem
Simple Explanation
From any point outside a circle, exactly two tangent lines can be drawn to the circle, and their two tangent segments — from the external point to each point of tangency — are always exactly equal in length.
Why Do We Need It?
This symmetry is a genuinely useful shortcut, instantly giving you a second known length whenever you find one tangent length from an external point.
See It
A circle with centre O, an external point P, and two tangent segments from P touching the circle at A and B, of equal length
Formula
The Two-Tangent Theorem
PA = PB, for two tangents from external point P touching the circle at A and B
The two tangent segments drawn from any single external point to a circle are always exactly equal in length.
- P
- — an external point (outside the circle)
- A, B
- — the two points where the tangents from P touch the circle
When to use it: Whenever two tangent segments are drawn to a circle from the same external point.
Worked Example
Use the two-tangent theorem
Two tangents are drawn from external point P to a circle, touching it at A and B. If PA = 14, find PB.
Why Does This Work?
Stated simply: triangles OAP and OBP are congruent by RHS (right angle at A and B from the tangent-radius theorem, equal hypotenuse OP shared by both, and equal radii OA=OB) — congruent triangles have all corresponding sides equal, so PA must equal PB.
Real-Life Example
Symmetric belt tension around a pulley
A belt or cable is pulled taut from a single anchor point to two tangent contact points on a circular pulley.
The two-tangent theorem guarantees both straight sections of belt from the anchor to the pulley are exactly the same length, a useful design certainty.
Practice
Two tangents from external point Q touch a circle at C and D. If QC = 3x+2 and QD = 5x−6, find x. (Using PA=PB.)
MediumCommon mistake
Applying the two-tangent theorem to tangents drawn from two DIFFERENT external points — the equal-length guarantee only applies to two tangents sharing the exact same external point.
Quick Review
- PA = PB, for tangents from the same external point P.
- Proof idea: congruent right triangles (RHS), using two equal radii and a shared hypotenuse.
- A quick way to find a second tangent length once one is known.