lies inside the triangle as stated above.) If the three vertices are located at , , and , and the sides opposite these vertices have corresponding lengths , , and , then the incenter is at Distances to vertices Denoting the incenter of triangle ABC as I, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation Additionally, where R and r are the triangle's circumradius and inradius respectively. Related constructions Other centers The distance from the incenter to the centroid is less than one third the length of the longest median of the triangle. By Euler's theorem in geometry, the squared distance from the incenter I to the circumcenter O is given by, p. 232. where R and r are the circumradius and the inradius respectively; thus the circumradius is at least twice the inradius, with equality only in the equilateral case. The distance from the incenter to the center N of the nine point circle is The squared distance from the incenter to the orthocenter H is Inequalities include: The incenter is the Nagel point of the medial triangle (the triangle whose vertices are the midpoints of the sides) and therefore lies inside this triangle. Conversely the Nagel point of any triangle is the incenter of its anticomplementary triangle. The incenter must lie in the interior of a disk whose diameter connects the centroid G and the orthocenter H (the orthocentroidal disk), but it cannot coincide with the nine-point center, whose position is fixed 1/4 of the way along the diameter (closer to G). Any other point within the orthocentroidal disk is the incenter of a unique triangle. Euler line The Euler line of a triangle is a line passing through its circumcenter, centroid, and orthocenter, among other points. The incenter generally does not lie on the Euler line; it is on the Euler line only for isosceles triangles, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers. Denoting the distance from the incenter to the Euler line as d, the length of the longest median as v, the length of the longest side as u, the circumradius as R, the length of the Euler line segment from the orthocenter to the circumcenter as e, and the semiperimeter as s, the following inequalities hold: Area and perimeter splitters Any line