Right triangle a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled "O", "A" and "H" respectively, then the trigonometric functions are For the expression of hyperbolic functions as ratio of the sides of a
Incircle and excircles of a triangle has area formula_39. Similarly, formula_40 has area formula_41 and formula_42 has area formula_43. Since these three triangles decompose formula_35, we see that the area formula_45 is: where formula_48 is the area of formula_35 and formula_50 is its semiperimeter. For an alternative formula, consider formula_51. This is a right-angled triangle with one side equal to "r" and the other side equal to formula_52. The same is true for formula_53. The large triangle is composed of 6 such triangles and the total area is: The Gergonne triangle (of "ABC") is defined by the 3 touchpoints of the incircle on the 3 sides.