An incenter is the center of an

incircle, which is a circle tangent to all three sides of a triangle.
The

trilinear coordinates
of this center is 1 : 1 : 1.

**Claim:** The incenter is located at the

concurrent point of the three

angle bisectors of a triangle.

**Proof:** (Draw along if it helps.) For triangle ABC,
let X be the

intersection of the

angle bisector *l*_{A} of vertex A, and the angle bisector

*l*_{B} of

vertex B.
From point X, construct

orthogonal lines to sides AB, BC, and CA, and call the orthogonal

intersection points as
C', A', and B', respectively.
Because of same angles and shared sides, the following

triangles form

congruent pairs:

AXB' ≅ AXC'

BXA' ≅ BXC'

CXA' ≅ CXB'

As such, A', B', and C' are

equidistant from X. Since they are

orthogonal intersections, they are the shortest distance from X to the sides. Hence a

circle I centered at X containing all points A', B', and C' tangentially on the

circumference is the

incircle.

Conversely, by examining in reverse the tangent points of any incircle, it becomes clear that the incenter must lie on all angle bisectors, thus proving uniqueness.