Fuhrmann circle

In geometry, the Fuhrmann circle of a triangle, named after the German Wilhelm Fuhrmann (1833–1904), is the circle with a diameter of 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]

Fuhrmann circle

Fuhrmann circle with Fuhrmann triangle (red),
Nagel point ${\displaystyle N}$ and orthocenter ${\displaystyle H}$
${\displaystyle |AP_{a}|=BP_{b}|=|CP_{c}|=2r}$

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]