Saccheri–Legendre theorem Saccheri–Legendre theorem In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid. The theorem is named after Giovanni Girolamo Saccheri and Adrien-Marie Legendre. The existence of at least one triangle with angle sum of 180 degrees in absolute geometry implies Euclid's parallel postulate. Similarly, the existence of at least one triangle with angle sum of less than 180 degrees
Incircle and excircles of a triangle Because the Incenter is the same distance of all sides the trilinear coordinates for the incenter are The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by where formula_7, formula_8, and formula_9 are the lengths of the sides of the triangle, or equivalently (using the law of sines) by where formula_11, formula_12, and formula_13 are the angles at the three vertices. The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices