Fuhrmann circle
Encyclopedia
In geometry
, the Fuhrmann circle of a triangle
is the circle
with a diameter
of the line segment
between the orthocenter and the Nagel point
.
The radius of the Fuhrmann circle of a triangle with sides a, b, and c and circumradius R is
which is also the distance between the circumcenter and incenter.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the Fuhrmann circle of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
is the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
with a diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of the line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
between the orthocenter and the Nagel point
Nagel point
In geometry, the Nagel point is a point associated with any triangle. Given a triangle ABC, let TA, TB, and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively...
.
The radius of the Fuhrmann circle of a triangle with sides a, b, and c and circumradius R is
which is also the distance between the circumcenter and incenter.