Umkreisberührungskreis

An Umkreisberührungskreis, also known as a tangential circle or excentral circle, is a circle related to a triangle that is tangent to one side of the triangle and the extensions of the other two sides. Every triangle has three such circles, one for each side. These circles are also referred to as excircles.

Each excircle is associated with a specific vertex of the triangle. The excircle opposite a vertex is tangent to the side of the triangle that does not contain that vertex, and it is also tangent to the lines containing the other two sides of the triangle. The center of an excircle is called an excenter. The excenters are the intersection points of the external bisectors of two angles of the triangle and the internal bisector of the third angle.

The radius of an excircle is called the exradius. There are formulas relating the exradius to the area of the triangle and its side lengths. For a triangle with sides a, b, and c, and area A, the exradii denoted by $r_a$, $r_b$, and $r_c$ (opposite vertices A, B, and C respectively) can be calculated as $r_a = A/(s-a)$, $r_b = A/(s-b)$, and $r_c = A/(s-c)$, where s is the semi-perimeter of the triangle (s = (a+b+c)/2). The excircles play a role in various geometric theorems and constructions involving triangles.