Fermat's right triangle theorem is, by the Pythagorean theorem, they would form two integer-sided right triangles in which the pair formula_10 gives one leg and the hypotenuse of the smaller triangle and the same pair also forms the two legs of the larger triangle. But if (as Fibonacci asserted) no square congruum can exist, then there can be no two integer right triangles that share two sides in this way. Because the congrua are exactly the numbers that are four times the area of a Pythagorean triangle, and multiplication by four does not change whether a number is square, the existence of a square
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