Fuhrmann circle

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.^[1]

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

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

which is also the distance between the circumcenter and incenter.^[2]

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.^[3]

Notes

^ 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).
^ Weisstein, Eric W. "Fuhrmann Circle". MathWorld.
^ Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, pp. 49-52

External links

Fuhrmann circle

[1] 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).

[2] Weisstein, Eric W. "Fuhrmann Circle". MathWorld.

[3] Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, pp. 49-52

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Fuhrmann circle

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