Foundations of geometry and therefore are valid theorems in absolute geometry. Proposition I.31 proves the existence of parallel lines (by construction). Also, the Saccheri–Legendre theorem, which states that the sum of the angles in a triangle is at most 180°, can be proved. The theorems of absolute geometry hold in hyperbolic geometry as well as in Euclidean geometry. Absolute geometry is inconsistent with elliptic geometry: in elliptic geometry there are no parallel lines at all, but in absolute geometry parallel lines do exist. Also, in elliptic geometry, the sum of the angles in any triangle is greater than 180°. Logically, the axioms do
Incircle and excircles of a triangle any vertex to the incircle tangency on either adjacent side is half the sum of the vertex's adjacent sides minus half the opposite side. Thus for example for vertex "B" and adjacent tangencies "T" and "T", The incircle radius is no greater than one-ninth the sum of the altitudes. The squared distance from the incenter "I" to the circumcenter "O" is given by and the distance from the incenter to the center "N" of the nine point circle is The incenter lies in the medial triangle (whose vertices are the midpoints of the sides). The radius of the incircle is