Angle Bisector Theorem: Concepts, Formulas and Examples

An angle bisector is a ray from a vertex that cuts an angle into two exactly equal halves. In a triangle, the three bisectors meet at the incenter — the center of the circle touching all three sides from inside.

The Angle Bisector Theorem connects angles with side lengths: the bisector of an angle divides the opposite side in the same ratio as the two sides forming that angle. This lets you find unknown lengths without measuring angles, and it appears constantly in school geometry and Olympiad problems.

Key Concepts

• Angle bisector: a ray from a vertex that divides an angle into two congruent (equal) angles.
• Internal bisector: the bisector inside the triangle; it meets the opposite side between its endpoints.
• Theorem statement: in triangle ABC, if AD bisects angle A and meets BC at D, then BD / DC = AB / AC.
• Notation: with a = BC, b = CA, c = AB, the theorem reads BD / DC = c / b.
• Ratios are dimensionless: BD and DC must use the same units, which cancel out.

Formulas

• Angle Bisector Theorem: BD / DC = AB / AC, assuming D lies on segment BC (internal bisector). It holds in every triangle.
• Segment lengths: BD = a × c / (b + c) and DC = a × b / (b + c), where a = BC — from the ratio plus BD + DC = a.
• Bisector length: if t is the bisector from A to BC, then t² = bc − BD × DC (internal bisector only).

Worked Example

In triangle ABC, AB = 6 cm, AC = 9 cm, and BC = 10 cm. The bisector of angle A meets BC at D. Find BD and DC.

1. Apply the theorem: BD / DC = AB / AC = 6 / 9 = 2 / 3.
2. Write BD = 2k and DC = 3k for some number k. Since D lies on BC, 2k + 3k = 10, so 5k = 10 and k = 2.
3. Therefore BD = 2 × 2 = 4 cm and DC = 3 × 2 = 6 cm.
4. Check: 4 / 6 = 2 / 3, which matches 6 / 9, and 4 + 6 = 10. Correct.

Common Misconceptions

• The bisector does not cut the opposite side in half. It divides the side in the ratio of the adjacent sides. It bisects the side only when the triangle is isosceles with AB = AC.
• Angle bisector, median, and altitude are different lines. A median joins a vertex to the midpoint of the opposite side, and an altitude is perpendicular to the opposite side. All three coincide only in an isosceles or equilateral triangle.
• The ratio uses the sides forming the bisected angle. Students sometimes write BD / DC = AB / BC by mistake. The ratio always involves the two sides that meet at the bisected vertex.

Real-World Applications

• Optics: the law of reflection splits the angle between incoming and outgoing rays like a bisector, used in mirror and lens design.
• Architecture: bisectors position supports symmetrically inside triangular roof trusses.
• Navigation: the incenter of a triangle of landmarks is equally distant from all three connecting roads — ideal for placing facilities like fire stations.

The Angle Bisector Theorem says the internal bisector of angle A divides BC in the ratio AB : AC. From this one ratio you can compute both segments of the split side, and even the bisector's own length. Remember: the bisector splits the angle, not the side.