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The angle bisector theorem

An angle bisector cuts the opposite side in the ratio of the other two sides — inside and outside the triangle — and every point on it is equidistant from the two arms.

High school Lesson 6 of 9 Interactive figure Practice

First, the rules

One ratio, inside and outside

★ Interior bisector theorem

Examples 1, 2, 3

|AB||AC| = |BD||DC|
AD bisects angle A and meets BC at D. The two pieces of BC are in the same ratio as the two sides leaving A.

Exterior bisector theorem

Example 1

|AB||AC| = |EB||EC|
AE bisects the exterior angle at A and meets line BC at E (outside the triangle). Same ratio, measured from E.

★ Equidistant from the arms

Example 2

A point that is the same distance from both arms of an angle lies on the bisector. Conversely, every point of the bisector is equidistant from the arms.

Splitting a segment in a ratio

Example 3

To split 16 in the ratio 3 : 5, count 3 + 5 = 8 parts of 16 / 8 = 2 each: the pieces are 6 and 10.

See it

Drag the corners — the ratio stays

The bisector from A is drawn to D on BC. Move any vertex and compare |AB| / |AC| with |BD| / |DC|: they stay equal. Switch to the exterior bisector to see the same ratio from E.

Step by step

Worked examples

Interior and exterior bisector together

both theorems

In triangle ABC, AD is the interior bisector at A with |BD| = 12 and |DC| = 8. AE is the exterior bisector at A, meeting line BC at E beyond C, with |CE| = x. Find x.

Ask yourself: Can the interior bisector tell you |AB| / |AC|? Does the same ratio hold for the exterior bisector? (Yes, both.)

Show the solution

Interior bisector: |AB||AC| = |BD||DC| = 128 = 32.
Exterior bisector gives the same ratio: |EB||EC| = 32.
|EB| = 12 + 8 + x = 20 + x, so 20 + xx = 32.
Cross-multiply: 40 + 2x = 3x.

Answer x = 40

A road map

equidistant = bisector

Town D is 18 km from A along road AB and is equally far from roads AB and AC. On road BC, |BD| = 6 km and |DC| = 9 km, where D is the junction on BC. |AB| = 18 km. Find |AC|.

Schematic, not to scale.

Ask yourself: Where is a point that is equally far from both arms of an angle? (On the bisector.)

Show the solution

D is equidistant from AB and AC, so AD is the bisector of angle A.
Bisector theorem: |AB||AC| = |BD||DC| ⇒ 18|AC| = 69.
6 · |AC| = 18 · 9 = 162

Answer |AC| = 27 km

Bisector and median on one side

ratio split

Right triangle ABC, right angle at B, |AB| = 12, |AC| = 20. AN bisects angle A, AD is the median to BC. Find |ND|.

Ask yourself: First get |BC| from the 12-16-20 triple, then split it in the ratio 12 : 20.

Show the solution

|BC| = 16 (3-4-5 scaled by 4).
|BN||NC| = 1220 = 35: 8 parts of 2, so |BN| = 6.
The median halves BC: |BD| = 8.
|ND| = 8 − 6

Answer |ND| = 2

Practice: the bisector ratio

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Three of the four lengths are given — find the fourth. Interior and exterior cases both appear. Your best score is kept in this browser only.

About this lesson

The angle bisector theorem: |AB| / |AC| = |BD| / |DC| for the interior bisector from A, and the same ratio for the exterior bisector. Drag the vertices and watch the ratio stay equal, see why a point equidistant from both arms lies on the bisector, follow worked examples and practise finding the segments.

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