Saccheri-Legendre Theorem

The existence of at least one triangle with angle sum of 180 degrees in absolute geometry implies Euclid's parallel postulate.

The theorem is named after Giovanni Girolamo Saccheri and Adrien-Marie Legendre.

The theorem is named after Giovanni Girolamo Saccheri and Adrien-Marie Legendre.

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.

It appeared in Saccheri's 1733 book Euclides ab omni naevo vindicatus [Euclid Freed of Every Flaw] but his work fell into obscurity.