Famous Theorems of Mathematics/Geometry

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Plane Euclidean Geometry[edit]

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[edit]

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

Projective Geometry[edit]

Elliptic Geometry[edit]

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[edit]

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.