In Euclidean geometry, Carnot's theorem, named after Lazare Carnot (1753–1823), is as follows. Let ABC be an arbitrary triangle. Then the sum of the signed distances from the circumcenter D to the sides of triangle ABC is
where r is the inradius and R is the circumradius. Here the sign of the distances is taken negative if and only if the line segment DX (X = F, G, H) lies completely outside the triangle. In the picture DF is negative and both DG and DH are positive.
Carnot's theorem is used in a proof of the Japanese theorem for concyclic polygons.