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,
Heronian triangle triangles with these sidelengths, and joins them together along the sides of length "a", to obtain a triangle with integer side lengths "c", "e", and "b" + "d", and with area If "a" is even then the area "A" is an integer. Less obviously, if "a" is odd then "A" is still an integer, as "b" and "d" must both be even, making "b"+"d" even too. Some Heronian triangles cannot be obtained by joining together two right-angled triangles with integer sides as described above. For example, a 5, 29, 30 Heronian triangle with area 72 cannot be constructed from two