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Carnot's Theorem


CarnotsTheorem

Given any triangle ABC, the signed sum of perpendicular distances from the circumcenter O to the sides (i.e., signed lengths of the pedal lines from O) is

 OO_A+OO_B+OO_C=R+r,

where r is the inradius and R is the circumradius. The sign of the distance is chosen to be negative iff the entire segment OO_i lies outside the triangle.

Explicitly,

 OO_A+OO_B+OO_C=(abc(|cosA|+|cosB|+|cosC|))/(4|Delta|),

where Delta is the area of triangle DeltaABC.


See also

Carnot's Polygon Theorem, Japanese Theorem

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References

Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, pp. 256 and 262, 1972.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 25, 1985.

Referenced on Wolfram|Alpha

Carnot's Theorem

Cite this as:

Weisstein, Eric W. "Carnot's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CarnotsTheorem.html

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