Overview

Angle bisectors turn angles into side ratios. You will need them for area and segment-ratio problems, and frequently appear in AMC 8, AMC 10, and basic olympiad geometry.

The most important result is the Angle Bisector Theorem, which changes angle information into side-length information. This makes many geometry problems simple algebra.

Definitions and Key Ideas

Consider triangle $ABC$, and let the bisector of $\angle A$ intersect side $BC$ at $D$.

The Angle Bisector Theorem states that

In words, an angle bisector divides the opposite side in the ratio of the adjacent sides.

Because triangles $ABD$ and $ACD$ share the same altitude from $A$, their areas satisfy

Combining this with the Angle Bisector Theorem gives

This is one of the fastest shortcuts for area-ratio problems.

Another useful fact is that every interior angle bisector meets the opposite side between its endpoints, dividing the triangle into two smaller triangles.

The Angle Bisector Theorem

If $AD$ bisects $\angle A$ in triangle $ABC$, then

This theorem is often used together with

to find unknown segment lengths.

For example, if

then

If $BC=14$, then

Area Ratios

One of the most common AMC applications of angle bisectors involves areas.

Since triangles $ABD$ and $ACD$ share the same altitude from $A$,

Using the Angle Bisector Theorem,

Thus, if

then

No area formulas are required.

Recognizing Angle Bisector Problems

Common clues that suggest using the Angle Bisector Theorem include:

• A segment is explicitly described as an angle bisector.
• Two side lengths adjacent to an angle are given.
• A ratio on the opposite side is requested.
• Area ratios appear alongside an angle bisector.
• A triangle is divided into two smaller triangles by a segment from a vertex.

Whenever you see an angle bisector and side lengths together, try the theorem immediately.

Common Pitfalls

• Using the theorem on a median instead of an angle bisector.
• Mixing up side lengths and segment lengths.
• Writing

instead of

• Forgetting to use

when the entire side length is known.

• Applying the theorem without first confirming that the segment is actually an angle bisector.

Worked Example

In triangle $ABC$, $AB=8$, $AC=6$, and $BC=7$. The bisector from $A$ meets $BC$ at $D$. Find $BD$.

Since

the segments are $4$ and $3$, so

More Examples

Example 1: Quick Ratio

In triangle $ABC$, $AB=5$ and $AC=10$. The angle bisector from $A$ meets $BC$ at $D$. Find $BD:DC$.

By the Angle Bisector Theorem,

Example 2: Area Ratio

If

find $BD:DC$.

The triangles share an altitude from $A$, so

Example 3: Finding Segments

In triangle $ABC$, $AB=12$, $AC=18$, and $BC=20$. The angle bisector from $A$ meets $BC$ at $D$. Find $DC$.

The Angle Bisector Theorem gives

Since

the side is divided into $5$ equal parts.

Each part has length $4$, so

Example 4: Area From Side Ratios

In triangle $DIH$, the angle bisector from $D$ meets $IH$ at $C$. Suppose

and

Find $[DIC]$.

Since

we have

The total area is $30$, so

Strategy Checklist

• Confirm the segment is an angle bisector.
• Immediately write

• Use

when a full side length is given.

• Convert area information into side ratios whenever possible.

• Look for shared altitudes when comparing triangle areas.

Practice Problems