In geometry, Euler's line (red line in the image), named after Leonhard Euler, is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine-point circle (red point) of any triangle. Image File history File links Triangle. ... Image File history File links Triangle. ... Table of Geometry, from the 1728 Cyclopaedia. ... Euler redirects here. ... In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... In geometry, the nine-point circle is a circle that can be constructed for any given triangle. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...
Leonhard Euler showed that in any triangle, those four points are collinear. The center of the nine-point circle lies midway between the orthocenter and the circumcenter, and the distance from the centroid to the circumcenter is half that from the centroid to the orthocenter. Euler redirects here. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...
Euler worked in an astonishing variety of areas, ranging from the very pure the theory of numbers, the geometry of a circle and musical harmony via such areas as infinite series, logarithms, the calculus and mechanics, to the practical optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides.
Euler proved, using his generating functions, that for any number, the number of odd partitions is always equal to the number of distinct partitions an intriguing and unexpected result.
Euler proved the pretty result that these three points all lie in a straight line now called the Eulerline of the triangle and that the centroid lies exactly one third of the distance between the other two.
Euler's paper arguably marks the beginning of topology and graph theory.
Euler added that, if the sum is one less that the noted number, the journey must begin from one of the areas marked with an asterisk, and it must begin from an unmarked one if the sum is equal.
Euler concluded that the desired journey can be made if it starts from area D or E. He then went on in his paper to develop simplified rules for determining whether a bridge-crossing problem has a solution.
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