Right triangle 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, falls on the midpoint of the hypotenuse. In any right triangle the diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times formula_44 In a right triangle with legs "a", "b" and hypotenuse "c", with equality only in the isosceles case. If the altitude from the hypotenuse is denoted "h",
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