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In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in . More precisely, let M be a surface in three-dimensional Euclidean space, and p a point on M.
Euler's partition theorem (number theory) Euler's polyhedron theorem ; Euler's quadrilateral theorem ; Euler's rotation theorem ; Euler's theorem (differential geometry) Euler's theorem (number theory) Euler's theorem in geometry (triangle geometry) Euler's theorem on homogeneous functions (multivariate calculus)
Euler's theorem states that in a triangle: = +, where and are the circumradius and inradius respectively, and is the distance between the circumcenter and the incenter. For excircles the equation is similar:
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
In geometry, the Euler line, named after Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər), is a line determined from any triangle that is not equilateral.It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
Euler's theorem Euler's theorem states that if n and a are coprime positive integers, then a φ(n) is congruent to 1 mod n. Euler's theorem generalizes Fermat's little theorem. Euler's totient function For a positive integer n, Euler's totient function of n, denoted φ(n), is the number of integers coprime to n between 1 and n inclusive.
Nine-point conic and Euler line generalization at Dynamic Geometry Sketches Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line. N J Wildberger. Chromogeometry. Discusses the nine-point circle with regard to three different quadratic forms (blue, red, green).