Integer triangle rational formula_14 where formula_15 reduces the generated Heronian triangle to its primitive and formula_16 scales up this primitive to the required size. A Pythagorean triangle is right angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad. All Pythagorean triples formula_17 with hypotenuse formula_18 which are primitive (the sides having no common factor) can be generated by where "m" and "n" are coprime integers and one of them is even with "m" > "n". Every even number greater than 2 can be the leg of a Pythagorean triangle (not necessarily primitive)
Automedian triangle much more recent. The problem of characterizing automedian triangles was posed in the late 19th century in the "Educational Times" (in French) by Joseph Jean Baptiste Neuberg, and solved there with the formula 2"a" = "b" + "c" by William John Greenstreet. Apart from the trivial cases of equilateral triangles, the triangle with side lengths 17, 13, and 7 is the smallest (by area or perimeter) automedian triangle with integer side lengths. There is only one automedian right triangle, the triangle with side lengths proportional to 1, , and . This triangle is the second triangle in the spiral of