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,
Fermat's right triangle theorem of a right triangle with square area. By dividing by any common factors, one can assume that this triangle is primitive and from the known form of all primitive Pythagorean triples, one can set formula_23, formula_24, and formula_25, by which the problem is transformed into finding relatively prime integers formula_26 and formula_27 (one of which is even) such that formula_28 is square. The four linear factors formula_26, formula_27, formula_31, and formula_32 are relatively prime and therefore must themselves be squares; let formula_33 and formula_34. Both formula_35 and formula_36 must be odd since exactly one of formula_26 or formula_27 is even