Carnot's theorem is the standard name for a circumcenter identity and is one of several distinct Carnot's
theorems in geometry (Honsberger 1985, Perrier 2007).
For any triangle 
, the signed sum of perpendicular distances from the circumcenter 
to the sides
is
 |
where 
is the inradius and 
is the circumradius. The
sign of the distance is chosen to be negative iff 
and the interior of the triangle
lie in opposite half-planes bounded by the corresponding
extended side.
Explicitly,
 |
where 
is the area of triangle 
. The signed distances
are 
,
, and 
, so an obtuse angle contributes
a negative term (Perrier 2007).
