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.

Name of Topic	Subtopics
Lines and Angles	Parallel Lines
Polygons	Triangles, Pythagorean Theorem, Quadrilaterals, Polygon Construction
Trigonometry	Basic Trigonometry, Sine and Cosine Rules, Trigonometrical Identities
Curves	Circles, Conic Sections
Coordinate Geometry	Plane Cartesian Coordinates, Plane Polar Coordinates, Coordinate Transformations

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.

Name of Topic	Subtopics
Lines, Planes and Angles
Polyhedra	Platonic Solids, Archimedean Solids
Curved Solids and Surfaces	Spheres, Cylinders, Cones, Quadrics

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.