Geometry • Topic 10

Incenter

The incenter ( $I$ ) is the center of the inscribed circle (incircle), which is tangent to all three sides internally.

Definition

Definition (Incenter)

The incenter is the intersection of the three internal angle bisectors. It is equidistant from the three sides (

r

A re a = r \cdot s

where

s = \frac{a + b + c}{2}

is the semi-perimeter.

∠ B I C = 9 0^{\circ} + \frac{A}{2}

This is a critical Olympiad lemma ("The Trillium Theorem"). Let the angle bisector of

A

intersect the circumcircle at

M

(the midpoint of arc

BC

). Then:

MB = MC = M I

M

is the center of a circle passing through

B, I, C

The incenter is the weighted average of vertices with side lengths as weights:

I = \frac{a A + b B + c C}{a + b + c}

Exercise (Problem 1)

In a right triangle with legs 3 and 4, find the inradius

r

Exercise (Problem 2)

△ A BC

∠ A = 6 0^{\circ}

. Prove that

I B = I C

is impossible unless the triangle is isosceles.

Exercise (Problem 3)

Use the "Fact 5" lemma to prove that the circumcenter of

△ B I C

lies on the circumcircle of

△ A BC