Which geometrical theorem is a fundamental relation in Euclidean geometry involving the longest side of a right-angled triangle?
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Hypotenuse 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.
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Pythagorean theorem 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":
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Non-Archimedean geometry In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry.
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Fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:
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Regular isotopy In the mathematical subject of knot theory, regular isotopy is the equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only. The notion of regular isotopy was introduced by Louis Kauffman (Kauffman 1990). It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of framed links, since the 2nd and 3rd Reidemeister moves preserve the winding number of the diagram (Kauffman 1990, pp. 450ff.). However, for diagrams in the sphere (considered as the plane plus infinity), the two notions are equivalent, due to the extra freedom of passing a strand through infinity.
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Euclidean plane isometry In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections (see below under classification of Euclidean plane isometries).
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Tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an "incircle"). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices.
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Krein–Milman theorem In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about convex sets in topological vector spaces. A particular case of this theorem, which can be easily visualized, states that given a convex polygon, one only needs the corners of the polygon to recover the polygon shape. The statement of the theorem is false if the polygon is not convex, as then there can be many ways of drawing a polygon having given points as corners.
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Fundamental theorem The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.
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Minkowski's second theorem In mathematics, Minkowski's second theorem is a result in the Geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.
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Bicentric quadrilateral In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called the "inradius" and "circumradius", and "incenter" and "circumcenter" respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also been called a double circle quadrilateral.
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Theorema Egregium Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface does not change if one bends the surface without stretching it. In other words, Gaussian curvature can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.
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Blaschke–Lebesgue theorem In plane geometry the Blaschke–Lebesgue theorem, named after Wilhelm Blaschke and Henri Lebesgue, states that the Reuleaux triangle has the least area of all curves of given constant width. By the isoperimetric inequality, the curve of constant width with the largest area is a circle.
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Coxeter decompositions of hyperbolic polygons A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other along that side. Hyperbolic polygons are the analogues of Euclidean polygons in hyperbolic geometry. A hyperbolic "n"-gon is an area bounded by "n" segments, rays, or entire straight lines. The standard model for this geometry is the Poincaré disk model. A major difference between Euclidean and hyperbolic polygons is that the sum of internal angles of a hyperbolic polygon is not the same as Euclidean polygons. In particular, the sum of the angles of a hyperbolic triangle is less than 180 degrees. Coxeter decompositions are named after Harold Scott MacDonald Coxeter, an accomplished 20th century geometer. He introduced the Coxeter group, an abstract group generated by reflections. These groups have many uses, including producing the rotations of Platonic solids and tessellating the plane.
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Fundamental theorem of curves In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size) completely determined by its curvature and torsion.
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Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle formula_1 points formula_2, formula_3, and formula_4 lie on segments formula_5, formula_6, and formula_7, then writing formula_8formula_9, formula_10formula_11, and formula_12formula_13, the signed area of the triangle formed by the cevians formula_14, formula_15, and formula_16 is the area of triangle formula_1 times
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Gallai–Hasse–Roy–Vitaver theorem In graph theory, the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations of its edges. It states that the minimum number of colors needed to properly color any graph "G" equals one plus the length of a longest path in an orientation of "G" chosen to minimize this path's length. The orientations for which the longest path has minimum length always include at least one acyclic orientation.
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Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. If "x" , "y" , and "z" are the lengths of the sides of the triangle, with no side being greater than "z" , then the triangle inequality states that
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Rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e. "handedness" of space). Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation.
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Bonnet theorem In the mathematical field of differential geometry, more precisely, the theory of surfaces in Euclidean space, the Bonnet theorem states that the first and second fundamental forms determine a surface in R uniquely up to a rigid motion. It was proven by Pierre Ossian Bonnet in about 1860.
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Brahmagupta theorem In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. It is named after the Indian mathematician Brahmagupta.
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Killing–Hopf theorem In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. These manifolds are called space forms. The Killing–Hopf theorem was proved by and .