The Fuhrmann triangle of a reference triangle is the triangle formed by reflecting the mid-arc points , , about the lines , , and .
The Fuhrmann triangle has trilinear vertex matrix
(1)
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The area of the Fuhrmann triangle is given by
(2)
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(3)
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where is the area of the reference triangle, is the distance between the circumcenter and incenter of the reference triangle, and is the circumradius of the reference triangle (P. Moses, pers. comm., Aug. 18, 2005).
The side lengths are
(4)
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(5)
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(6)
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The circumcircle of the Fuhrmann triangle is called the Fuhrmann circle, and the lines , , and concur at the circumcenter .
Surprisingly, the orthocenter of the Fuhrmann triangle is the incenter of the reference triangle. Furthermore, the nine-point center of the Fuhrmann triangle and are coincident, and the radius of the nine point circle of the Fuhrmann triangle is (P. Moses, pers. comm., Aug. 18, 2005).
The following table gives the centers of the Fuhrmann triangle in terms of the centers of the reference triangle that correspond to Kimberling centers .