: Mathématiques: Carnot's theorem

geometry > Euclidean geometry > Carnot's theorem

Carnot's theorem

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
${\\displaystyle DF+DG+DH=R+r,\\ }$
where r is the inradius and R is the circumradius of the triangle. Here the sign of the distances is taken 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Carnot%27s_theorem_(inradius,_circumradius))

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