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

D
F
+
D
G
+
D
H
=
R
+
r
,
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.