In Euclidean geometry, Carnot's theorem (English: kar-NOH, French: [kaʁno]) states that the sum of the signed distances from the circumcenter
D
{\displaystyle D}
to the sides of an arbitrary triangle
A
B
C
{\displaystyle ABC}
is
D
F
+
D
G
+
D
H
=
R
+
r
,
{\displaystyle DF+DG+DH=R+r,\ }
where
r
{\displaystyle r}
is the inradius and
R
{\displaystyle 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
D
X
{\displaystyle DX}
(for
X
{\displaystyle X}
equal to
F
{\displaystyle F}
,
G
{\displaystyle G}
, or
H
{\displaystyle H}
) lies completely outside the triangle. For example, in the diagram,
D
F
{\displaystyle DF}
is negative and both
D
G
{\displaystyle DG}
and
D
H
{\displaystyle DH}
are positive. The theorem can be viewed as saying that the sum of the actual-distance trilinear coordinates of the circumcenter is
R
+
r
{\displaystyle R+r}
.
The theorem is named after Lazare Carnot (1753–1823). It is used in a proof of the Japanese theorem for concyclic polygons.