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Angle bisector theorem

2026-09-06Plane GeometryMs Ho

A bisector divides the opposite side in the ratio of the adjacent sides.

Key formulas

BD/DC = AB/AC

Statement

If AD bisects ∠A and meets BC at D, then BD/DC = AB/AC.

Example

AB=8, AC=6, BC=7. Then BD/DC=8/6=4/3. With BD+DC=7, BD=4, DC=3.

Proof idea

One standard proof uses area ratios or the law of sines in triangles ABD and ACD.

Worked example

Worked example

Problem. In △ABC, AB = 8, AC = 6, BC = 7. The bisector of ∠A meets BC at D. Find BD and DC.

Solution. BD/DC = AB/AC = 8/6 = 4/3. With BD + DC = 7: BD = 4, DC = 3.

Not altitude

The bisector is not generally perpendicular to BC.

Review quiz

AB=10, AC=15, BC=20. Bisector meets BC at D. BD equals:

Practice questions

  1. AB=10, AC=15, BC=20. Find BD and DC.
  2. State the converse of the angle bisector theorem.

Study notes

Internal only?

There is also an external bisector version with a minus sign in the ratio.

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