Famous Theorems of Mathematics/Geometry

Plane Euclidean Geometry
Euclidean geometry is the form of geometry defined and studied by Euclid. It is generally distinguished from non-Euclidean geometries by the parallel postulate, which (in Euclid's formulation) states "that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles".

This section covers theorems that relate to Euclidean geometry in two dimensions.

Solid Euclidean Geometry
This section covers theorems that relate to Euclidean geometry in three dimensions. Many proofs in three-dimensional geometry rely on results in plane geometry.

Elliptic Geometry
Elliptic geometry is a non-Euclidean geometry in which there are no parallel straight lines – any coplanar straight lines will intersect if sufficiently extended. The surface of a sphere, considered as a geometric space in its own right, exhibits this kind of geometry.

Hyperbolic Geometry
Hyperbolic geometry is a non-Euclidean geometry in which every straight line has a continuum of parallel straight lines (in the 'never meeting' sense) through the same point.