Triangle inequality ≥ b ≥ c" and "formula_8" is the golden ratio, as In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides, and less than their sum. The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle . An isosceles triangle is constructed with equal sides . From the triangle postulate, the angles in the right triangle satisfy: Likewise, in the isosceles triangle , the
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