Fuhrmann circle

In geometry, the Fuhrmann circle of a triangle is the circle with a diameter of the line segment between the orthocenter H and the Nagel point N.

The radius of the Fuhrmann circle of a triangle with sides a, b, and c and circumradius R is

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

which is also the distance between the circumcenter and incenter.

See also

• Nagel point
• Incircle and excircles of a triangle