Carnot's Theorem

Given any Triangle $A_1A_2A_3$, the signed sum of Perpendicular distances from the Circumcenter $O$ to the sides is

$$OO_1+OO_2+OO_3=R+r,$$

where $r$ is the Inradius and $R$ is the Circumradius. The sign of the distance is chosen to be Positive Iff the entire segment $OO_i$ lies outside the Triangle.

See also Japanese Triangulation Theorem

References

Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, pp. 256 and 262, 1972.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 25, 1985.