Question: Which geometrical theorem is a fundamental relation in Euclidean geometry involving the longest side of a right-angled triangle?

Context articles: Hilbert's axioms
Foundations of geometry
Euclidean space
Clifford parallel
Pythagorean theorem
Generalized trigonometry
Birkhoff's axioms
Pythagoras Peak
Non-Euclidean geometry
Hypotenuse

Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book "Grundlagen der Geometrie" (tr.
 "The Foundations of Geometry") as the foundation for a modern treatment of Euclidean geometry.
 Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
Foundations of geometry is the study of geometries as axiomatic systems.
 There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries.
 These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint.
 The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view.
 The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
 It is named after the Ancient Greek mathematician Euclid of Alexandria.
 The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry.
 Euclidean spaces also generalize to higher dimensions.
In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point.
 The concept was first studied by William Kingdon Clifford in elliptic space.
 Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, but in fact the "lines" of elliptic geometry are curves, and they have finite length, unlike the lines of Euclidean geometry.
 The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.
In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
 It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
 The theorem can be written as an equation relating the lengths of the sides "a", "b" and "c", often called the "Pythagorean equation":
Ordinary trigonometry studies triangles in the Euclidean plane R.
 There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, definitions using functional equations.
 Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry.
 Generally, trigonometry can be the study of triples of points in any kind of geometry or space.
 A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices.
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms.
 These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor.
 Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry.
Pythagoras Peak is the highest peak, 1,275 m, in the central Tula Mountains, standing along the north side of Beaver Glacier, 8 nautical miles (15 km) southeast of Mount Storer.
 The peak has a prominent notch, the eastern aspect being a right-angled triangle with a perpendicular northern face.
 It was photographed from Mount Riiser-Larsen in February 1958 by ANARE (Australian National Antarctic Research Expeditions) led by Phillip Law, but was first visited and surveyed in December 1958 by G.A. Knuckey, ANARE surveyor.
 It was named by the Antarctic Names Committee of Australia (ANCA) after Pythagoras, Greek philosopher, whose theorem concerning a right-angled triangle is well known.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.
 As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one.
 In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.
 When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.
In geometry, a hypotenuse (rarely: "hypothenuse") is the longest side of a right-angled triangle, the side opposite of the right angle.
 The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.