Equilateral triangle terms of side length "a" can be derived directly using the Pythagorean theorem or using trigonometry. The area of a triangle is half of one side "a" times the height "h" from that side: The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base "a", and the hypotenuse is the side "a" of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem so that Substituting "h" into the area formula (1/2)"ah" gives the area formula for the equilateral triangle: Using trigonometry, the area of
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