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,
Medial triangle of "ABC": the parallel to "AB" through "C", the parallel to "AC" through "B", and the parallel to "BC" through "A". Trilinear coordinates for the vertices of the anticomplementary triangle, X'Y'Z', are given by The name "anticomplementary triangle" corresponds to the fact that its vertices are the anticomplements of the vertices A, B, C of the reference triangle. The vertices of the medial triangle are the complements of A, B, C. Medial triangle Each side of the medial triangle is called a "midsegment" (or "midline"). In general, a midsegment of a triangle is a line segment which joins the midpoints