The Fuhrmann triangle, named after Wilhelm Fuhrmann (1833–1904), is special triangle based on a given arbitrary triangle.

Fuhrmann triangle (red):
mid arc points:
Fuhrmann triangle (red):

For a given triangle and its circumcircle the midpoints of the arcs over triangle sides are denoted by . Those midpoints get reflected at the associated triangle sides yielding the points , which forms the Fuhrmann triangle.[1][2]

The circumcircle of Fuhrmann triangle is the Fuhrmann circle. Furthermore the Furhmann triangle is similar to the triangle formed by the mid arc points, that is .[1] For the area of the Fuhrmann triangle the following formula holds:[3]

Where denotes the circumcenter of the given triangle and its radius as well as denoting the incenter and its radius. Due to Euler's theorem one also has . The following equations hold for the sides of the Fuhrmann triangle:[3]

Where denote the sides of the given triangle and the sides of the Fuhrmann triangle (see drawing).

References

edit
  1. ^ a b 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. ^ Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, pp. 49-52
  3. ^ a b Weisstein, Eric W. "Fuhrmann triangle". MathWorld. (retrieved 2019-11-12)