Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c is negative if the segment joining the circumcenter to the side does not pass through the interior of the triangle, and it is positive otherwise.

Then we have the following elegant result:

Carnot's theorem. a+b+c=R+r

Move the vertices of the triangle to see this equality in action.

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Created by David Richeson using GeoGebra.