Fuhrmann Triangle

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

For a given triangle $\triangle ABC$ and its circumcircle the midpoints of the arcs over triangle sides are denoted by $M_a, M_b, M_c$. Those midpoints get reflected at the associated triangle sides yielding the points $M^\prime_a, M^\prime_b, M^\prime_c$, which forms the ''Fuhrmann triangle''. Roger A. Johnson: ''Advanced Euclidean Geometry''. Dover 2007, , pp. 228–229, 300 (originally published 1929 with Houghton Mifflin Company (Boston) as ''Modern Geometry'').Ross Honsberger: ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry''. MAA, 1995, pp
49-52

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 $\triangle M^\prime_c M^\prime_b M^\prime_a \sim \triangle M_a M_b M_c$. For the area of the Fuhrmann triangle the following formula holds: (retrieved 2019-11-12)

:$, \triangle M^\prime_c M^\prime_b M^\prime_a, = \frac=\frac$

Where $O$ denotes the circumcenter of the given triangle $\triangle ABC$ and $R$ its radius as well as $I$ denoting the incenter and $r$ its radius. Due to Euler's theorem one also has $, OI, ^2=R(R-2r)$. The following equations hold for the sides of the Fuhrmann triangle:

:$a^\prime=\sqrt, OI,$
:$b^\prime=\sqrt, OI,$
:$c^\prime=\sqrt{\frac{(a+b-c)(a+b+c)}{ab, OI,$

Where $a, b, c$ denote the sides of the given triangle $\triangle ABC$ and $a^\prime, b^\prime, c^\prime$ the sides of the Fuhrmann triangle (see drawing).

References

Objects defined for a triangle