The name Carnot's theorem is used for several distinct results in geometry. Their statements involve different configurations and should be distinguished as follows.
The circumcenter form, commonly called Carnot's theorem, states that the signed distances from the
circumcenter of a triangle
to its sides sum to 
, where 
and 
are the circumradius and
inradius (Honsberger 1985).
Carnot's perpendicularity theorem characterizes when three perpendiculars to the extended sides of a triangle are concurrent by an identity between sums of squared distances (Bogomolny).
Carnot's conic theorem characterizes when six points, two on each extended side of a triangle, lie on a conic section. A cyclic product of six signed ratios equals 1 (Szilasi 2012).
Carnot's polygon theorem gives a product-of-ratios identity for a plane cutting
a skew quadrilateral. Its planar algebraic
curve version relates the intersections of an
algebraic curve of polynomial
degree 
with the extended sides of a polygon.
For a triangle, it includes Menelaus'
theorem and Carnot's conic theorem as
the line and conic section
cases (Altshiller-Court 1979, Szilasi 2012).
The law of cosines, 
, is also called Carnot's theorem in Italian
usage (University of Bologna).
Carnot's equal-angle extension of the Simson line theorem states that, for a point on the circumcircle of a triangle, three lines through that point making the same directed angle with the cyclically oriented sides meet those extended sides in three collinear points. The Simson line is the perpendicular case (Tran Minh Ngoc 2018, Theorem 3.6).
