Fuhrmann circle

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File:Fuhrmann circle.svg

File:Fuhrmann circle triangle.svg

In geometry, the Fuhrmann circle of a triangle, named after the German Wilhelm Fuhrmann (1833–1904), is the circle that has as a diameter the line segment between the orthocenter $H$ and the Nagel point $N$. This circle is identical with the circumcircle of the Fuhrmann triangle. Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, {{ISBN|978-0-486-46237-0}}, pp. 228–229, 300 (originally published 1929 with Houghton Mifflin Company (Boston) as Modern Geometry).

The radius of the Fuhrmann circle of a triangle with sides a, b, and c and circumradius R is

: $R\backslash sqrt\{\backslash frac\{a^3-a^2b-ab^2+b^3-a^2c+3abc-b^2c-ac^2+c^3\}\{abc\}\},$

which is also the distance between the circumcenter and incenter.{{mathworld|urlname=FuhrmannCircle|title=Fuhrmann Circle}}

Aside from the orthocenter the Fuhrmann circle intersects each altitude of the triangle in one additional point. Those points all have the distance $2r$ from their associated vertices of the triangle. Here $r$ denotes the radius of the triangles incircle.Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, pp. [https://books.google.com/books?id=7vcrmWEV73AC&pg=PA49 49-52]