is using the grassfire transform, in which one forms a continuous sequence of offset curves, each at some fixed distance from the polygon; the medial axis is traced out by the vertices of these curves. In the case of a triangle, the medial axis consists of three segments of the angle bisectors, connecting the vertices of the triangle to the incenter, which is the unique point on the innermost offset curve. The straight skeleton, defined in a similar way from a different type of offset curve, coincides with the medial axis for convex polygons and so also has its junction at the incenter. Proofs Ratio proof Let the bisection of and meet at , and the bisection of and meet at , and and meet at . And let and meet at . Then we have to prove that is the bisection of . In , , by the Angle bisector theorem. In , . Therefore, , so that . So is the bisection of Perpendicular proof A line that is an angle bisector is equidistant from both of its lines when measuring by the perpendicular. At the point where two bisectors intersect, this point is perpendicularly equidistant from the final angle's forming lines (because they are the same distance from this angles opposite edge), and therefore lies on its angle bisector line. Relation to triangle sides and vertices Trilinear coordinates The trilinear coordinates for a point in the triangle give the ratio of distances to the triangle sides. Trilinear coordinates for the incenter are given by The collection of triangle centers may be given the structure of a group under coordinatewise multiplication of trilinear coordinates; in this group, the incenter forms the identity element. Barycentric coordinates The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by where , , and are the lengths of the sides of the triangle, or equivalently (using the law of sines) by where , , and are the angles at the three vertices. Cartesian coordinates The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter