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,
Integer triangle "c" , "a" + "b" ≤ "c" and "a" ≤ "b" ≤ "c". This is also the number of integer sided obtuse or right (non-acute) triangles with largest side "c". The sequence starting at "c" = 1, is: Consequently, the difference between the two above sequences gives the number of acute integer sided triangles (up to congruence) with given largest side "c". The sequence starting at "c" = 1, is: By Heron's formula, if "T" is the area of a triangle whose sides have lengths "a", "b", and "c" then Since all the terms under the radical on the right