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 ${\displaystyle H}$ and the Nagel point ${\displaystyle N}$. 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

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

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 ${\displaystyle 2r}$ from their associated vertices of the triangle. Here ${\displaystyle r}$ denotes the radius of the triangles incircle.^[3]

Notes

Further reading

• Nguyen Thanh Dung: "The Feuerbach Point and the Fuhrmann Triangle". Forum Geometricorum, Volume 16 (2016), pp. 299–311.
• J. A. Scott: An Eight-Point Circle. In: The Mathematical Gazette, Volume 86, No. 506 (Jul., 2002), pp. 326–328 (JSTOR)

External links

• Fuhrmann circle

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This page was last edited on 13 February 2024, at 15:19