Carnot's theorem (inradius, circumradius)

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D
G
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D
H
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F
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D
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D
H
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{\displaystyle {\begin{aligned}&DG+DH+DF\\={}&|DG|+|DH|+|DF|\\={}&R+r\end{aligned}}} Carnot theorem2.svg

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

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

The theorem is named after Lazare Carnot (17531823). It is used in a proof of the Japanese theorem for concyclic polygons.

References