mittenpunkt

The mittenpunkt is a geometric concept related to triangles. It is also known as the tangential triangle's circumcenter or the circumcenter of the tangential triangle. The mittenpunkt has various interesting properties and connections to other triangle centers. It lies on the Euler line of the triangle. Specifically, the mittenpunkt is the circumcenter of the triangle formed by the points where the excircles of a triangle are tangent to its sides.

The barycentric coordinates of the mittenpunkt are given by (a(b+c-a) : b(c+a-b) : c(a+b-c)). This formula highlights its relationship with the side lengths of the triangle. The mittenpunkt can be constructed using compass and straightedge, though the process can be somewhat involved. It is not as commonly discussed as more prominent triangle centers like the centroid or incenter, but it holds a unique position within the geometry of a triangle. The term "mittenpunkt" is German, roughly translating to "middle point," which hints at its central nature within the context of the tangential triangle.