incenter

The incenter of a triangle is the point where the three internal angle bisectors intersect. It is the center of the incircle, the circle tangent to all three sides of the triangle.

The incenter is equidistant from the three sides; this common distance is called the inradius, denoted r. The incircle touches each side at a point where the perpendicular from the incenter meets that side. Consequently, the incenter lies inside the triangle for every nondegenerate triangle.

Key formulas and coordinates: If a, b, c are the side lengths opposite vertices A, B, C, and s = (a + b + c)/2 is the semiperimeter, then the area Δ satisfies Δ = r s, so r = Δ / s. In barycentric coordinates relative to triangle ABC, the incenter has coordinates a : b : c. In Cartesian coordinates, the incenter can be written as I = (aA + bB + cC) / (a + b + c), where A, B, C are the position vectors of the vertices.

Touchpoints and lengths: The incircle touches sides BC, CA, and AB at D, E, and F, respectively. On BC, for example, the segments satisfy BD = s − b and DC = s − c; analogous relations hold on the other sides. These distances are derived from the equal tangents from the triangle’s vertices to the incircle.

The incenter has several notable properties, including its role as the center of the unique circle inscribed in the triangle (the incircle) and its relationship to the triangle’s area and perimeter.