Exterior Angle Bisector — Definition, Formula & Examples

An exterior angle bisector is the ray that divides an exterior angle of a triangle (or polygon) into two equal parts. It lies outside the interior of the original angle and is perpendicular to the interior angle bisector of the same vertex.

Given a triangle $ABC$ , the exterior angle bisector at vertex $A$ is the locus of points in the plane of the triangle that bisects the angle supplementary to $\angle BAC$ . By the external angle bisector theorem, this line meets line $BC$ at a point $D$ such that $\dfrac{BD}{DC} = \dfrac{AB}{AC}$ , where $D$ divides segment $BC$ externally.

Key Formula

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

Where:

$D$ = Point where the exterior bisector at vertex A meets line BC (external division)
$AB$ = Length of the side from A to B
$AC$ = Length of the side from A to C

How It Works

At any vertex of a triangle, extending one side past the vertex creates an exterior angle. The exterior angle bisector splits that exterior angle into two congruent angles. A key property is the external angle bisector theorem: the bisector meets the line containing the opposite side at a point that divides it externally in the ratio of the two adjacent sides. The interior and exterior angle bisectors at the same vertex are always perpendicular to each other.

Worked Example

Problem: In triangle ABC, AB = 6, AC = 4, and B and C lie on a line. The exterior angle bisector at vertex A meets line BC at point D. Find BD and DC if BC = 5, with D dividing BC externally.

Step 1: Apply the external angle bisector theorem. Point D divides BC externally in the ratio AB : AC = 6 : 4 = 3 : 2.

\frac{BD}{DC} = \frac{3}{2}

Step 2: For external division, D is on the extension of BC beyond C. Using the external division formula: BD = BD and DC = BD − BC. Set BD/DC = 3/2.

\frac{BD}{BD - 5} = \frac{3}{2}

Step 3: Cross-multiply and solve: 2·BD = 3·(BD − 5), so 2BD = 3BD − 15, giving BD = 15 and DC = 10.

BD = 15, \quad DC = 10

Answer: D lies on line BC extended beyond C, with BD = 15 and DC = 10. The ratio 15 : 10 = 3 : 2 confirms the theorem.

Why It Matters

The exterior angle bisector is essential in advanced triangle geometry. The three exterior angle bisectors determine the excenters of a triangle — the centers of the excircles, which appear in competition mathematics and in calculating a triangle's area via its inradius and exradii.

Common Mistakes

Mistake: Confusing the external angle bisector theorem with the internal angle bisector theorem, and using internal division instead of external division.

Correction: The interior bisector divides the opposite side internally in the ratio of the adjacent sides, while the exterior bisector divides it externally in the same ratio. External division means the point lies outside segment BC.