GeoProof — Angle Bisector Theorem explorer

a (BC) b (AC) c (AB)

BD / DC = AB / AC

The dashed bisector splits angle A into two equal halves — and lands on BC at exactly the point that matches the side ratio.

The Angle Bisector Theorem

In a triangle ABC, draw the bisector of the angle at A — the ray that splits ∠A into two equal halves. It meets the opposite side BC at a point D. The theorem says D divides BC in the same ratio as the two sides that form angle A:

BD / DC = AB / AC

So the bisector always lands closer to the shorter of the two adjacent sides.

What to watch

Drag a vertex to make AB and AC very different in length and D slides toward the shorter side; make AB = AC (an isosceles triangle) and D lands exactly at the midpoint of BC, since the ratio becomes 1.

Why it's true

One classic proof draws a line through C parallel to the bisector AD, meeting line AB extended at E. That makes triangle ACE isosceles with AE = AC, and similar triangles give BD/DC = AB/AE = AB/AC. An area argument works too: triangles ABD and ACD share the same height from A, so their areas — and hence the bases BD and DC — are in the ratio of AB to AC.

The converse holds as well

If a point D on BC satisfies BD/DC = AB/AC, then AD must be the bisector of angle A. The condition and the bisector are equivalent.

Where it's used

It underlies the incenter (where all three bisectors meet), constructions for dividing a segment in a given ratio, and many olympiad-style length-chasing problems.

A bit of history

The result is essentially Proposition 3 of Book VI of Euclid's Elements (c. 300 BCE), built on the theory of proportions and similar triangles developed there.