Euler line the Euler line is parallel to an acute triangle's side "BC" if and only if formula_43 The locus of the centroids of equilateral triangles inscribed in a given triangle is formed by two lines perpendicular to the given triangle's Euler line. In a right triangle, the Euler line coincides with the median to the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides,
Incircle and excircles of a triangle 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 lies inside the triangle as stated above.) If the three vertices are located at formula_14, formula_15, and formula_16, and the sides opposite these vertices have corresponding lengths formula_7, formula_8, and formula_9, then the incenter is at 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