Angle Bisectors & the Incenter

An angle bisector is a ray that divides an angle into two congruent angles. It carries its own equidistance fact, but this one is about sides, not endpoints: every point on an angle bisector is the same perpendicular distance from the two sides of the angle. If $\overrightarrow{BD}$ bisects $\angle ABC$ and $P$ sits on that bisector, then the perpendicular from $P$ to $\overrightarrow{BA}$ and the perpendicular from $P$ to $\overrightarrow{BC}$ have equal lengths.

Bisect all three angles of a triangle and the three bisectors meet at one point, the incenter. Because the incenter is on every bisector at once, it is equidistant from all three sides of the triangle — it is the center of the circle that fits snugly inside the triangle. This lesson has two problem types: half-angle arithmetic and equal-distance equations. Both are quick once you know which fact to reach for.

The angle bisector theorem

The theorem says: if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. The distance from a point to a line always means the perpendicular distance — the length of the segment that meets the side at a right angle. In figures, those perpendiculars are usually drawn dashed with right-angle marks.

So when a problem gives you expressions for the two perpendicular distances, like $PE = 4x + 1$ and $PF = 7x - 14$ , the theorem lets you set them equal and solve. As always, reread the question at the end — it usually asks for a length, so substitute $x$ back in.

Half-angle arithmetic

The other standard problem uses the definition directly. A bisector cuts an angle into two congruent halves, so each half is half of the whole: if $\overrightarrow{BD}$ bisects $\angle ABC$ , then $\displaystyle m\angle ABD = m\angle DBC = \frac{1}{2}\, m\angle ABC$ .

Read carefully which piece you are given. If you know a half, double it to get the whole angle. If you know the whole angle, halve it to get each piece. Mixing those two directions is the most common error on this skill.

The incenter

The three angle bisectors of a triangle are concurrent at the incenter. The incenter is equidistant from the three sides of the triangle, and that shared perpendicular distance is the radius of the inscribed circle — the circle inside the triangle that touches all three sides.

Unlike the circumcenter, the incenter is always inside the triangle, for every triangle type. And keep the distances straight: the incenter is equidistant from the sides, not from the vertices. The distances $IX$ , $IY$ , $IZ$ to the corners are generally three different lengths.

In the figure below, the three angle bisectors meet at the incenter $I$ , and the circle centered at $I$ touches all three sides — its radius is the shared perpendicular distance.

Worked examples

Example 1: equal distances to the sides

$\overrightarrow{BD}$ bisects $\angle ABC$ . Point $P$ on the bisector has perpendicular distances $PE = 2x + 5$ and $PF = 4x - 9$ to the two sides. Find $PE$ .

A point on the bisector is equidistant from the sides of the angle

PE = PF

Set the expressions equal

2x + 5 = 4x - 9

Solve for

x

14 = 2x, \text{ so } x = 7

Substitute back into

PE

PE = 2(7) + 5 = 19

Answer: $PE = 19$

Example 2: from a half to the whole angle

$\overrightarrow{BD}$ bisects $\angle ABC$ and $m\angle DBC = 31^\circ$ . Find $m\angle ABC$ .

A bisector makes two congruent halves

m\angle ABD = m\angle DBC = 31^\circ

Add the halves

m\angle ABC = 31^\circ + 31^\circ

Simplify

m\angle ABC = 62^\circ

Answer: $m\angle ABC = 62^\circ$

Example 3: distances from the incenter

$I$ is the incenter of $\triangle XYZ$ , and the perpendicular distance from $I$ to side $\overline{XY}$ is $6$ . What is the perpendicular distance from $I$ to each of the other two sides?

The incenter is equidistant from all three sides

\text{equal perpendicular distances}

Use the given distance

6

That distance is also the radius of the inscribed circle

r = 6

Answer: $6$ units to each side

Try one yourself

Common questions

What is the difference between the incenter and the circumcenter?

The incenter comes from the angle bisectors and is equidistant from the three sides — it is the center of the inscribed circle. The circumcenter comes from the perpendicular bisectors of the sides and is equidistant from the three vertices — it is the center of the circumscribed circle.

Is the distance measured to the vertex or to the side?

To the side, and always along a perpendicular. The angle bisector theorem compares the perpendicular distances from a point to the two sides of the angle. It says nothing about distances to vertices.

Can the incenter ever be outside the triangle?

No. Every angle bisector of a triangle stays inside the triangle, so their meeting point does too. The incenter is inside for acute, right, and obtuse triangles alike.

How is this different from a perpendicular bisector?

An angle bisector splits an angle and belongs to a vertex; a perpendicular bisector splits a segment and belongs to a side. Their equidistance facts are different too: angle bisector points are equidistant from two sides, perpendicular bisector points are equidistant from two endpoints.

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