Carnot's theorem
For the similarly named theorem in thermodynamics, see Carnot's theorem (thermodynamics).
In Euclidean geometry, Carnot's theorem, named after Lazare Carnot (1753–1823), is as follows. Let ABC be an arbitrary triangle. Then the sum of the signed distances from the circumcenter D to the sides of triangle ABC is
where r is the inradius and R is the circumradius. 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.
Carnot's theorem is used in a proof of the Japanese theorem for concyclic polygons.
External links
- Weisstein, Eric W., "Carnot's theorem", MathWorld.
- Carnot's Theorem at cut-the-knot
- Yet another Carnot's Theorem with multiple applications at cut-the-knot
- Carnot's Theorem by Chris Boucher. The Wolfram Demonstrations Project.
