teorema de Carnot

EN Texto em inglês Ainda não há artigo no seu idioma: trecho em inglês.

In Euclidean geometry, Carnot's theorem (English: kar-NOH, French: [kaʁno]) states that the sum of the signed distances from the circumcenter D {\displaystyle D} to the sides of an arbitrary triangle A B C {\displaystyle ABC} is D F + D G + D H = R + r , {\displaystyle DF+DG+DH=R+r,\ } where r {\displaystyle r} is the inradius and R {\displaystyle 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 D X {\displaystyle DX} (for X {\displaystyle X} equal to F {\displaystyle F} , G {\displaystyle G} , or H {\displaystyle H} ) lies completely outside the triangle. For example, in the diagram, D F {\displaystyle DF} is negative and both D G {\displaystyle DG} and D H {\displaystyle DH} are positive. The theorem can be viewed as saying that the sum of the actual-distance trilinear coordinates of the circumcenter is R + r {\displaystyle R+r} . The theorem is named after Lazare Carnot (1753–1823). It is used in a proof of the Japanese theorem for concyclic polygons.