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In Euclidean geometry, Carnot's theorem, named after Lazare Carnot (1753–1823), is as follows. Let ABC be an arbitrary triangle. Then the sum of the signed distances from the circumcenter D to the sides of triangle ABC is

\( DF + DG + DH = R + r,\ \)

where r is the inradius and R is the circumradius. Here the sign of the distances is taken negative if and only if the line segment DX (X = F, G, H) lies completely outside the triangle. In the picture DF is negative and both DG and DH are positive.

Carnot's theorem is used in a proof of the Japanese theorem for concyclic polygons.
External links

Weisstein, Eric W., "Carnot's theorem" from MathWorld.
Carnot's Theorem at cut-the-knot
Yet another Carnot's Theorem with multiple applications at cut-the-knot
Carnot's Theorem by Chris Boucher. The Wolfram Demonstrations Project.

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