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",
Incircle and excircles of a triangle The touchpoint opposite A is denoted "T", etc. This Gergonne triangle "TTT" is also known as the contact triangle or intouch triangle of "ABC". Its area is where formula_56, formula_57, formula_58 are the area, radius of the incircle and semiperimeter of the original triangle, and formula_7, formula_8, formula_9 are the side lengths of the original triangle. This is the same area as that of the extouch triangle. The three lines "AT", "BT" and "CT" intersect in a single point called the Gergonne point, denoted as "Ge" - "X(7)". The Gergonne point lies in the open orthocentroidal disk punctured at its