The circumcircle of the Fuhrmann triangle. It has the line , where is the orthocenter and Na is the Nagel point, as its diameter. In fact, these points (Kimberling centers and , respectively), are the only Kimberling centers lying on it.
While and are the only noteworthy triangle centers to lie on the circle, a total of at least six other noteworthy points lie on the Fuhrmann circle (Honsberger 1995, p. 49). Three of these are the points , , and which are a distance along the altitudes from the vertices, where is the inradius of (Honsberger 1995, p. 52).
The Fuhrmann circle has circle function
(1)
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corresponding to Kimberling center for an appropriate choice of the circle constant . Its center is known as the Fuhrmann center , and is Kimberling center . Its radius is
(2)
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(3)
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where is the distance between the circumcenter and incenter (P. Moses, pers. comm., May 9, 2005) and is the circumradius of the reference triangle.
Interestingly, the segment is parallel to the diameter of the Fuhrmann circle (P. Moses, pers. comm., May 9, 2005). Even more amazingly, the two parallelograms and have as their centroids the Spieker center and nine-point center, respectively, where or (F. M. Jackson, pers. comm., Apr. 26, 2007).