Carnot's theorem

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For the similarly named theorem in thermodynamics, see Carnot's theorem (thermodynamics).
\begin{align} & {} \qquad DG + DH + DF  \\ & {} = |DG| + |DH|- |DF| \\ & {} = R + r  \end{align}

In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter D to the sides of an arbitrary triangle ABC is

DF + DG + DH = R + r,\

where r is the inradius and R is the circumradius of the triangle. 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.

The theorem is named after Lazare Carnot (1753–1823). It is used in a proof of the Japanese theorem for concyclic polygons.

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