Carnot's theorem (inradius, circumradius)

In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter D to the sides of an arbitrary triangle ABC is
where r is the inradius and R is the circumradius of the triangle. Here the sign of the distances is taken to be negative if and only if the open line segment DX lies completely outside the triangle. For example, in the diagram, DF is negative and both DG and DH are positive. The theorem can be viewed as saying that the sum of the actual-distance trilinear coordinates of the circumcenter is.
The theorem is named after Lazare Carnot. It is used in a proof of the Japanese theorem for concyclic polygons.