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Thus, the nine-point center forms the center of a point reflection that maps the medial triangle to the Euler triangle, and vice versa. [citation needed] According to Lester's theorem, the nine-point center lies on a common circle with three other points: the two Fermat points and the circumcenter. [9] The Kosnita point of a triangle, a ...
The nine-point center N is one-fourth of the way along the Euler line from the centroid G to the orthocenter H: [6]: p.153 ¯ = ¯. Let ω be the nine-point circle of the diagonal triangle of a cyclic quadrilateral. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle.
Then f is a triangle center function and α : β : γ is the corresponding triangle center whenever the sides of the reference triangle are labelled so that a < b < c. Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest.
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.
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. This resource is hosted at the University of Evansville . It started from a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles by Professor Clark Kimberling .
Their point of concurrence is known as the triangle's Kosnita point (named by Rigby in 1997). It is the isogonal conjugate of the nine-point center . [ 3 ] [ 4 ] It is triangle center X ( 54 ) {\displaystyle X(54)} in Clark Kimberling's list . [ 5 ]
The incircle of a triangle ABC is a circle that is tangent to all three sides of the triangle. Its center, the incenter of the triangle, lies at the point where the three internal angle bisectors of the triangle cross each other. The nine-point circle is another circle defined from a triangle.
In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurrency of all three of the triangle's splitters.