Angle Bisector Theorem and Triangle Problems

The angle bisector theorem states that the bisector of an angle of a triangle divides the opposite side into two parts whose lengths are in the same ratio as the two adjacent sides. In brief: in triangle ABC, if AD bisects angle A with D on BC, then BD/DC = AB/AC. For AB = 6, AC = 9 and BC = 10, the point D splits BC into BD = 4 and DC = 6.

The theorem is a useful tool for finding lengths in a triangle without trigonometry, and it explains the location of the incenter. This guide states the theorem and its converse, gives a proof with similar triangles, covers the exterior bisector, includes a formula for the length of the bisector and works through typical triangle problems that combine the theorem with other facts.

What does the angle bisector theorem say?

In triangle ABC let the bisector of angle A meet side BC at D. Then

BD/DC = AB/AC.

The longer side adjacent to the angle gets the longer part of the opposite side. The converse is also true: if a point D on BC satisfies BD/DC = AB/AC, then AD bisects angle A. Using the sides a = BC, b = CA, c = AB, the segments are BD = ac/(b + c) and DC = ab/(b + c).

Example 1

AB = 6, AC = 9, BC = 10. Then BD/DC = 6/9 = 2/3. The parts are BD = 10 · 2/5 = 4 and DC = 10 · 3/5 = 6. Check: 4/6 = 2/3 ✓.

How do you prove it?

Draw a line through C parallel to AD, meeting the extension of BA at E. Since AD ∥ EC, the angle ∠AEC equals ∠BAD (corresponding angles) and ∠ACE equals ∠CAD (alternate angles). The bisector gives ∠BAD = ∠CAD, so ∠AEC = ∠ACE and triangle ACE is isosceles with AE = AC. By the intercept theorem in triangle BCE (with AD ∥ EC).

How long is the angle bisector?

The length of the bisector from A to D is given by the formula l_a = √(bc[(b + c)² − a²])/(b + c). Equivalently, AD² = bc − BD · DC.

Example 2

With b = 9, c = 6, a = 10: BD · DC = 4 · 6 = 24 and bc = 54, so AD² = 54 − 24 = 30 and AD = √30 ≈ 5.48. Using the other formula: √(54 · (225 − 100))/15 = √(54 · 125)/15 = √6750/15 = 82.16/15 ≈ 5.48 ✓.

What about the exterior angle bisector?

The bisector of the exterior angle at A meets the line BC at a point E outside the segment, and it divides BC externally in the same ratio: EB/EC = AB/AC (when AB ≠ AC). The internal and external bisectors at A are perpendicular. The points D and E, the foot points of the internal and external bisectors, are the ends of a diameter of the circle of Apollonius, the locus of points whose distances to B and C are in the ratio AB : AC.

Example 3

With AB = 6, AC = 9, BC = 10: for the external bisector, EB/EC = 2/3 with E outside, beyond B. Let EB = x, then EC = x + 10 and x/(x + 10) = 2/3, so 3x = 2x + 20 and x = 20. So E is 20 units from B.

How do you solve triangle problems with the theorem?

Example 4: finding sides

In triangle ABC, AB = 12, BC = 14 and the bisector from A meets BC at D with BD = 6. Then DC = 8 and AC = AB · DC/BD = 12 · 8/6 = 16. Check the triangle inequality: 12 + 14 > 16 ✓.

Example 5: the incenter divides the bisector

The incenter I lies on the bisector AD and divides it in the ratio AI : ID = (b + c) : a. For the triangle with a = 10, b = 9, c = 6: AI : ID = 15 : 10 = 3 : 2. So AI = (3/5) · 5.48 ≈ 3.29.

Example 6: an isosceles triangle

If AB = AC, the ratio is 1 and D is the midpoint of BC. The bisector is also the median and the altitude.

Example 7: a right triangle

In a right triangle with legs 3 and 4 and hypotenuse 5, the bisector of the right angle (at C) meets AB at D. Then AD/DB = AC/BC. Let AC = 3 and BC = 4: AD = 5 · 3/7 = 15/7 ≈ 2.14 and DB = 20/7 ≈ 2.86.

Worked values for sample triangles

In the last row BD = 9 · 7/15 = 4.2 and DC = 9 · 8/15 = 4.8.

Where does the theorem come from?

The theorem appears in Euclid’s Elements in Book VI (Proposition 3), stated in terms of proportions: if an angle of a triangle is bisected and the bisector cuts the base, the segments of the base have the same ratio as the remaining sides of the triangle. The formula for the length of the bisector is found in later works on geometry, and the result about the circle of Apollonius is named after Apollonius of Perga.

Where is it used today?

The theorem helps in geometry competitions and in computing positions in a triangle, for example the position of the incenter from the vertex coordinates: I = (a A + b B + c C)/(a + b + c). The same ratio appears in optics, in Snell’s-law problems with weights, and in mesh generation, where triangles are refined by bisecting angles.

Which mistakes are common?

Inverting the ratio

BD is next to AB, so BD/DC = AB/AC. The longer adjacent side corresponds to the longer part.

Confusing the bisector with the median

The median divides the opposite side in the ratio 1 : 1. The bisector does this only in an isosceles triangle with AB = AC.

Applying the theorem to the wrong side

The segments are on the side opposite the bisected angle. Check which vertex is the starting point.

Forgetting the triangle inequality

After computing a side length, check that the triangle exists.