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,
Isosceles triangle formula_10. When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral. If the two equal sides have length formula_2 and the other side has length formula_3, then the internal angle bisector formula_22 from one of the two equal-angled vertices satisfies as well as and conversely, if the latter condition holds, an isosceles triangle parametrized by formula_2 and formula_22 exists. The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by C. L. Lehmus. Its other namesake, Jakob Steiner, was one of the first