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:
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.