Fuhrmann circle

Fuhrmann circle
Fuhrmann circle with Fuhrmann triangle (red),
Nagel point N {\displaystyle N} and orthocenter H {\displaystyle H}
| A P a | = B P b | = | C P c | = 2 r {\displaystyle |AP_{a}|=BP_{b}|=|CP_{c}|=2r}

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

R a 3 − a 2 b − a b 2 + b 3 − a 2 c + 3 a b c − b 2 c − a c 2 + c 3 a b c , {\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 2 r {\displaystyle 2r} from their associated vertices of the triangle. Here r {\displaystyle r} denotes the radius of the triangles incircle.[3]

Notes

  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

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)
  • Fuhrmann circle
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