The Angle Bisector Theorem

An angle bisector of a triangle splits one of its angles into two equal halves and meets the opposite side. It divides that opposite side in a very predictable way.

Theorem The Angle Bisector Theorem

If AD bisects ∠BAC, then D divides the opposite side so that BD : DC equals AB : AC.

\frac{BD}{DC} = \frac{AB}{AC}

Example Finding a length

In triangle ABC, AB = 6, AC = 14, and BC = 18. AD bisects ∠BAC. Find BD.

\dfrac{BD}{DC} = \dfrac{AB}{AC} = \dfrac{6}{14} = \dfrac{3}{7}

Since BD + DC = 18:

BD = 18 \times \dfrac{3}{10}

⟹ BD = 5.4 and DC = 12.6.

Example Finding a side

In triangle PQR, PQ = 9. The bisector from P meets QR at S, with QS = 6 and SR = 9. Find PR.

\dfrac{PQ}{PR} = \dfrac{QS}{SR} = \dfrac{6}{9}

PR = 9 \times \dfrac{9}{6}

⟹ PR = 13.5.

Summary

An angle bisector splits the opposite side in the ratio of the two adjacent sides: BD : DC = AB : AC.
Always remember that BD + DC = BC — the two parts make up the whole side.
The three angle bisectors of a triangle meet at one point, the incenter (centre of the inscribed circle).
Common slip: don't mix up the two adjacent sides with the opposite side.