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 lA of vertex A, and the angle bisector lB 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:


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.