Carnot's Theorem

Carnot's theorem states that the signed sum of the perpendicular distances from the circumcenter of a triangle to the three sides is equal to the sum of the circumradius and the inradius of the triangle. What is meant by "signed sum" is that the distance is taken to be negative if the segment lies entirely outside the triangle. In the image, the blue segments are the perpendicular distances on the left-hand side of this equality, and the total length of the red segments is the right-hand side of this equality.