for all Euclidean triangles. Medians, angle bisectors, perpendicular side bisectors, and altitudes The medians and the sides are related by and , and equivalently for mb and mc. For angle A opposite side a, the length of the internal angle bisector is given by for semiperimeter s, where the bisector length is measured from the vertex to where it meets the opposite side. The interior perpendicular bisectors are given by where the sides are and the area is The altitude from, for example, the side of length a is Circumradius and inradius The following formulas involve the circumradius R and the inradius r: where ha etc. are the altitudes to the subscripted sides; and . The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle: Adjacent triangles Suppose two adjacent but non-overlapping triangles share the same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths (a, b, f) and (c, d, f), with the two triangles together forming a cyclic quadrilateral with side lengths in sequence (a, b, c, d). Then Centroid Let G be the centroid of a triangle with vertices A, B, and C, and let P be any interior point. Then the distances between the points are related by The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices: Let qa, qb, and qc be the distances from the centroid to the sides of lengths a, b, and c. Then and for area T. Circumcenter, incenter, and orthocenter Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius. Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by Lie algebras, that otherwise have the same properties as usual triangles. Euler's theorem states that the distance d between the circumcenter and the incenter is given by or equivalently where R is the circumradius and r is the inradius. Thus for all triangles R ≥ 2r, with equality