Carnot's theorem (inradius, circumradius)

$|DG|+|DH|-|DF|=R+r$

In Euclidean geometry, Carnot's theorem (Template:IPAc-en Template:Respell, Template:IPA) 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 to be negative if and only if the open line segment $DX$ (for $X$ equal to $F$, $G$, or $H$) lies completely outside the triangle. For example, in the diagram, $DF$ is negative and both $DG$ and $DH$ are positive. The theorem can be viewed as saying that the sum of the actual-distance trilinear coordinates of the circumcenter is $R+r$.

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

References

• Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, Template:ISBN, p.99
• Frédéric Perrier: Carnot's Theorem in Trigonometric Disguise. The Mathematical Gazette, Volume 91, No. 520 (March, 2007), pp. 115–117 (JSTOR)
• David Richeson: The Japanese Theorem for Nonconvex Polygons – Carnot's Theorem. Convergence, December 2013

External links

• Template:MathWorld
• Carnot's Theorem at cut-the-knot
• Carnot's Theorem by Chris Boucher. The Wolfram Demonstrations Project.