Saccheri–Legendre theorem implies the characteristic postulate of hyperbolic geometry. Max Dehn gave an example of a non-Legendrian geometry where the angle sum of a triangle is greater than 180 degrees, and a semi-Euclidean geometry where there is a triangle with an angle sum of 180 degrees but Euclid's parallel postulate fails. In Dehn's geometries the Archimedean axiom does not hold. 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
Heptagonal triangle heptagonal triangle's orthic triangle, with vertices at the feet of the altitudes, is similar to the heptagonal triangle, with similarity ratio 1:2. The heptagonal triangle is the only obtuse triangle that is similar to its orthic triangle (the equilateral triangle being the only acute one). The various trigonometric identities associated with the heptagonal triangle include these:, The cubic equation has solutions formula_66 and formula_67 which are the squared sines of the angles of the triangle. The positive solution of the cubic equation equals formula_69 which is twice the cosine of one of the triangle’s angles. We also have: For an