Lesson 6.1.3

Angle Bisectors & Incenter

An angle bisector splits an angle into two equal halves. In a triangle, the three angle bisectors meet at the incenter — the center of the inscribed circle.

Introduction

While the circumcenter uses perpendicular bisectors of sides, the incenter uses bisectors of angles. The incenter is always inside the triangle and marks the center of the largest circle that fits inside — the inscribed circle (or incircle).

Past Knowledge

Angle bisectors (1.3). Perpendicular bisectors & circumcenter (6.1.2).

Today's Goal

Use the Angle Bisector Theorem. Locate the incenter and inradius.

Future Success

Medians & centroid (6.1.4). Circle inscriptions (Ch 10).

Key Concepts

Angle Bisector Theorem

If a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle (measured as perpendicular distances). The converse is also true.

Incenter

The incenter is the intersection of a triangle's three angle bisectors.

It is equidistant from all three sides (not vertices)
It is the center of the inscribed circle that is tangent to all three sides
It is always inside the triangle (unlike circumcenter)
The inradius is the perpendicular distance from incenter to any side

Worked Examples

Basic

Using the Angle Bisector Theorem

Point lies on the bisector of . The distance from to is 7 cm. What is the distance from to ?

By the Angle Bisector Theorem, is equidistant from both sides.

Distance = 7 cm

Intermediate

Incenter Distances

In , incenter has (perpendicular to ). Find the distance from to and .

The incenter is equidistant from all three sides.

Both distances = 5 (inradius = 5)

Advanced

Incircle Area

A triangle has an inradius of and a semi-perimeter of . What is the triangle's area?

Area formula with inradius:

Area = 60 square units

Common Pitfalls

Confusing Incenter and Circumcenter

Incenter = angle bisectors, equidistant from sides. Circumcenter = perpendicular bisectors, equidistant from vertices.

Distance to Side ≠ Distance to Vertex

The inradius is the perpendicular distance to each side, not the distance to each vertex. These are very different measurements.

Real-Life Applications

Sprinkler Placement

To place a circular sprinkler that reaches all three edges of a triangular garden, place it at the incenter. The incircle is the largest circle that fits inside, maximizing coverage within the boundaries.

Logo Design — Inscribed Circles

Graphic designers use incircles to fit the largest possible circular element (logo, icon) inside triangular frames, ensuring it is tangent to all three edges for a balanced look.