Given a triangle △ABC, let T_{A}, T_{B}, T_{C} be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and the C-excircle meets line AB, respectively. The lines AT_{A}, BT_{B}, CT_{C}concur in the Nagel point N of triangle △ABC.

Another construction of the point T_{A} is to start at A and trace around triangle △ABChalf its perimeter, and similarly for T_{B} and T_{C}. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments AT_{A}, BT_{B}, CT_{C} are called the triangle's splitters.

There exists an easy construction of the Nagel point. Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge. We obtain three lines which concur at the Nagel point.^{[1]}

The un-normalized barycentric coordinates of the Nagel point are $(s-a:s-b:s-c)$ where $s={\tfrac {a+b+c}{2))$ is the semi-perimeter of the reference triangle △ABC.

^Gallatly, William (1913). The Modern Geometry of the Triangle (2nd ed.). London: Hodgson. p. 20.

^Baptist, Peter (1987). "Historische Anmerkungen zu Gergonne- und Nagel-Punkt". Sudhoffs Archiv für Geschichte der Medizin und der Naturwissenschaften. 71 (2): 230–233. MR0936136.