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
Fermat's right triangle theorem congruum is equivalent to the existence of a Pythagorean triangle with a square area. It is this variant of the problem that Fermat's proof concerns: he shows that there is no such triangle. In considering this problem, Fermat was inspired not by Fibonacci but by an edition of Diophantus published by Claude Gaspard Bachet de Méziriac. This book described various special right triangles whose areas had forms related to squares, but did not consider the case of areas that were themselves square. By rearranging the equations for the two Pythagorean triangles above, and then multiplying them together, one obtains the