Fuhrmann circle

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

${\displaystyle R{\sqrt {\frac {a^{3}-a^{2}b-ab^{2}+b^{3}-a^{2}c+3abc-b^{2}c-ac^{2}+c^{3}}{abc}}},}$

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 ${\displaystyle 2r}$ from their associated vertices of the triangle. Here ${\displaystyle r}$ denotes the radius of the triangles incircle.^[3]