Here are the axioms of Euclidean Geometry Any two points can be joined by a straight line. Any straight line segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. All right angles are congruent. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. From these, all the rich and interesting theorems of school geometry can be derived.

Euclidean Geometry is the geometry of the flat plane.

Euclid's Elements is a systematic study of the theorems and propositions that can be proven from the axioms.

For centuries it was thought that Euclidean Geometry was somehow "The" geometry, and that the axioms were self-evidently true in our world.

Nikolai Lobachevsky (1792 - 1856) and János Bolyai (1802 - 1860) proved independently that there were models of the first four axioms that did not satisfy the fifth, thus showing that the fifth postulate cannot be proven from the other four.

By using alternative versions of the fifth postulate we obtain so-called Non-Euclidean Geometry.

Some of this material is duplicated on the page about Axioms, from which it could perhaps be removed. ALternatively, an overview page on Geometry could be written.