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Topics referred to by the same term
Euclidean theorem may refer to: Any theorem in Euclidean geometry Any theorem in Euclid's Elements, and in particular: Euclid's theorem that there are
Euclidean_theorem
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
Length of a line segment
In mathematics, the Euclidean distance between two points in a Euclidean space is the length of the line segment between them. It can be calculated from
Euclidean_distance
Mathematical model of the physical space
other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on a Euclidean plane. Although many
Euclidean_geometry
Theorem in projective geometry
line of the hexagon. It is named after Blaise Pascal. The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with
Pascal's_theorem
On the remainder of division by x – r
algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It
Polynomial_remainder_theorem
Division with remainder of integers
restricted to integers, Euclidean division and the division theorem can be generalized to univariate polynomials over a field and to Euclidean domains. In the
Euclidean_division
Two geometries based on axioms closely related to those specifying Euclidean geometry
mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry
Non-Euclidean_geometry
Theorem in topology
one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance
Brouwer_fixed-point_theorem
Existence of a line through two points
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the
Sylvester–Gallai_theorem
Theorem about right triangles
In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle
Geometric_mean_theorem
About simultaneous modular congruences
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then
Chinese_remainder_theorem
Algorithm for computing greatest common divisors
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers
Euclidean_algorithm
Theorem in mathematical analysis
follows from the version for Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since
Sard's_theorem
Theorem in projective geometry
"complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet. This results in a projective plane. Desargues's theorem is true
Desargues's_theorem
Exterior angle of a triangle is greater than either of the remote interior angles
of axioms for Euclidean geometry is used (see Foundations of geometry) this assertion of Euclid can be proved. The exterior angle theorem is not valid
Exterior_angle_theorem
In absolute geometry, the sum of the angles in a triangle is at most 180°
axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid. The theorem is named after Giovanni
Saccheri–Legendre_theorem
theorem (Euclidean geometry) Butterfly theorem (Euclidean geometry) CPCTC (triangle geometry) Carnot's theorem (geometry) Casey's theorem (Euclidean geometry)
List_of_theorems
Every rigid motion is a screw displacement
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement
Chasles'_theorem_(kinematics)
Fixed-point theorem for set-valued functions
subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization
Kakutani_fixed-point_theorem
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Theorem relating the diameter of a point set to the minimum radius of an enclosing ball
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball
Jung's_theorem
Geometric model of the physical space
)\cdot d\mathbf {r} .} Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral
Three-dimensional_space
Gives a lower bound on the number of lines determined by n points in a projective plane
noted in their paper that the analogous (Euclidean) result is a consequence of the Sylvester–Gallai theorem, by an induction on the number of points.
De Bruijn–Erdős theorem (incidence geometry)
De_Bruijn–Erdős_theorem_(incidence_geometry)
Fundamental space of geometry
space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces
Euclidean_space
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R
Bolzano–Weierstrass_theorem
Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices
Ptolemy's_theorem
On distances from opposite corners to a point inside a rectangle
In Euclidean geometry, the British flag theorem says that if a point P is chosen inside a rectangle ABCD then the sum of the squares of the Euclidean distances
British_flag_theorem
Complete manifolds of non-negative sectional curvature largely reduce to the compact case
For this reason, the theorem is often stated only for non-compact manifolds. As a very simple example, take M to be Euclidean space Rn. The sectional
Soul_theorem
Every Riemannian manifold can be isometrically embedded into some Euclidean space
three-dimensional Euclidean space because curves drawn on the page retain the same arc length however the page is bent. The first theorem is for continuously
Nash_embedding_theorems
Theorem about the Euclidean algorithm
Lamé's Theorem is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm. Using Fibonacci numbers, he proved in 1844 that
Lamé's_theorem
Theorem in geometry
In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon
Varignon's_theorem
Theorem in differential topology
The hairy ball theorem of algebraic topology (formally, the Sphere Vector Field Theory, sometimes called the hedgehog theorem) states that there is no
Hairy_ball_theorem
Equality of triangles between three squares
In mathematics, specifically geometry, Cross's theorem, also known as Vecten's theorem, equates the area of a triangle to the area of each of the triangles
Cross's_theorem
Non-associative algebras with positive-definite quadratic form
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz, published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Relates the length of a median of a triangle to the lengths of its sides
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the
Apollonius's_theorem
On topological spaces where the intersection of countably many dense open sets is dense
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient
Baire_category_theorem
Theorem in set-theoretic geometry
The Banach–Tarski paradox is a theorem in set-theoretic geometry that states the following: Given a solid ball in three-dimensional space, there exists
Banach–Tarski_paradox
One of several theorems in different areas of mathematics
mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur
Schur's_theorem
Theorem about circles
In Euclidean geometry, the butterfly theorem is a classical result which can be stated as follows: Let M be the midpoint of a chord PQ of a circle, through
Butterfly_theorem
17th-century conjecture proved by Andrew Wiles in 1994
using the Euclidean algorithm (c. 5th century BC). Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point
Fermat's_Last_Theorem
Geometry theorem relating the line segments created by intersecting chords in a circle
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created
Intersecting_chords_theorem
Theorem in differential topology
differential topology Nash embedding theorem – Every Riemannian manifold can be isometrically embedded into some Euclidean spacePages displaying short descriptions
Whitney_embedding_theorem
Commutative ring with a Euclidean division
it is a Euclidean domain. Every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic:
Euclidean_domain
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) Take a small neighborhood of M in that Euclidean space, Nε. Extend the vector
Poincaré–Hopf_theorem
Theorem in vector calculus
Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of
Stokes'_theorem
Branch of mathematics
("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This
Geometry
Geometric relation on line segments formed by a line cutting through a triangle
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle
Menelaus's_theorem
Property of geometry, also used to generalize the notion of "distance" in metric spaces
of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms):
Triangle_inequality
Subset of Euclidean space is compact if and only if it is closed and bounded
mathematics, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S {\displaystyle S} of Euclidean space R n {\displaystyle
Heine–Borel_theorem
Shape with three sides
Theorem". American Mathematical Monthly. 115 (4): 330–338. doi:10.1080/00029890.2008.11920532. King, James R. (2021). Geometry Transformed: Euclidean
Triangle
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
proof of the theorem via supersymmetric Euclidean field theories was also found. The Chern–Gauss–Bonnet theorem can be seen as a special instance in the
Chern–Gauss–Bonnet_theorem
Relationship between derivatives and integrals
Euclidean Proof of the Fundamental Theorem of Calculus at Convergence Isaac Barrow's proof of the Fundamental Theorem of Calculus Fundamental Theorem
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a
Fubini's_theorem
Geometric object that has length and direction
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric
Euclidean_vector
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Mathematical transformation that preserves distances
(also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between
Rigid_transformation
Polynomial zeros related to linear factors
( X ) {\displaystyle f(X)} . The theorem may be proved using Euclidean division of polynomials: Perform a Euclidean division of f ( x ) {\displaystyle
Factor_theorem
Geometry theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that if A , B , C {\displaystyle A,B,C} is one set of collinear points
Pappus's_hexagon_theorem
In mathematics, a statement that has been proven
mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
Conditions under which a chaotic system can be reconstructed by observation
dimension dA. Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with k > 2 d A . {\displaystyle k>2d_{A}.}
Takens's_theorem
Theorem in Euclidean geometry
triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry. With the additional construction of
Japanese theorem for cyclic polygons
Japanese_theorem_for_cyclic_polygons
Integers have unique prime factorizations
The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains, Euclidean domains
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Geometry theorem relating line segments created by a secant and tangent line
In Euclidean geometry, the tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle
Tangent–secant_theorem
On four non-intersecting circles that lie inside a bigger circle and tangent to it
In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician
Casey's_theorem
Overview of and topical guide to geometry
Convex hull Coxeter group Euclidean distance Homothetic center Hyperplane Lattice Ehrhart polynomial Leech lattice Minkowski's theorem Packing Sphere packing
Outline_of_geometry
Counting polynomial roots in an interval
Sturm used the negative of the remainder of the Euclidean division of the two preceding ones. The theorem remains true if one replaces the negative of the
Sturm's_theorem
Result about 4 points on a line which cannot be derived from Euclid's postulates
c.] David Hilbert originally included Pasch's theorem as an axiom in his modern treatment of Euclidean geometry in The Foundations of Geometry (1899)
Pasch's_theorem
In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was
Busemann's_theorem
On the existence of hyperplanes separating disjoint convex sets
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar
Hyperplane_separation_theorem
Ancient Greek mathematician (fl. 300 BC)
the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories
Euclid
Theorem in Euclidean geometry
In Euclidean geometry, Anne's theorem describes an equality of certain areas within a convex quadrilateral. This theorem is named after the French mathematician
Anne's_theorem
Movement with a fixed point is rotation
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the body remains
Euler's_rotation_theorem
Theorems that help decompose a finite group based on prime factors of its order
specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow
Sylow_theorems
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
Branch of mathematics
are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development
Differential_geometry
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
On triangles inscribed in a circle with a diameter as an edge
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle
Thales's_theorem
Mathematical treatise by Euclid
These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many
Euclid's_Elements
Geometric model of the planar projection of the physical universe
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}
Euclidean_plane
Unit-distance-preserving maps are isometries
geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances
Beckman–Quarles_theorem
Topics referred to by the same term
theory Hammersley–Clifford theorem in probability Clifford's circle theorems in Euclidean geometry This disambiguation page lists mathematics articles associated
Clifford's_theorem
Theorem in topology about homeomorphic subsets of Euclidean space
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . It states: If U {\displaystyle
Invariance_of_domain
Every triangle with two angle bisectors of equal lengths is isosceles
S2CID 256110198. Weisstein, Eric W. "Steiner–Lehmus theorem". MathWorld. Paul Yiu: Euclidean Geometry Notes, Lectures Notes, Florida Atlantic University
Steiner–Lehmus_theorem
Sufficiently long sequences of numbers have long monotonic subsequences
In mathematics, the Erdős–Szekeres theorem asserts that, given r {\displaystyle r} and s {\displaystyle s} , any sequence of distinct real numbers with
Erdős–Szekeres_theorem
Method for computing the relation of two integers with their greatest common divisor
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest
Extended_Euclidean_algorithm
A prime p divides a^p–a for any integer a
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In
Fermat's_little_theorem
Fundamental theorem in mathematical logic
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability
Gödel's_completeness_theorem
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Theorem that any three objects in space can be simultaneously bisected by a plane
positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each
Ham_sandwich_theorem
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Theorem about orthocenter and polars in circle geometry
Brokard's theorem (also known as Brocard's theorem) is a theorem on poles and polars in projective geometry commonly used in Olympiad mathematics. It is
Brokard's_theorem
Theorem concerning ratios of line segments
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry
Intercept_theorem
Isometry group of Euclidean space
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations
Euclidean_group
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Theorem in measure theory
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures
Prokhorov's_theorem
Type of mathematical space
space has these properties. For compact subsets of Euclidean space, this is the extreme value theorem. Another basic property of finite sets is that every
Compact_space
Geometry without the parallel postulate
giving rise to Euclidean or hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry. However
Absolute_geometry
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Theorem in plane geometry
such pair has a unique intersection point in the extended Euclidean plane. Monge's theorem states that the three such points given by the three pairs
Monge's_theorem
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EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
EUCLIDEAN THEOREM
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