Root system inner product is the Killing form. The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer. This is because formula_32 and formula_33 are both integers, by assumption, and formula_34 Since formula_35, the only possible values for formula_36 are formula_37 and formula_38, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of "α" other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2"α" or −2"α", are
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",