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.
The incenter is located at the concurrent
point of the three angle bisectors
of a triangle.
(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
From point X, construct orthogonal
lines to sides AB, BC, and CA, and call the orthogonal intersection
C', A', and B', respectively.
Because of same angles and shared sides, the following triangle
s form congruent
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.