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 (X = F, G, H) lies completely outside the triangle.

In the diagram, DF is negative and both DG and DH are positive.

The theorem is named after Lazare Carnot (1753–1823).

It is used in a proof of the Japanese theorem for concyclic polygons.