Routh's theorem

Routh's theorem

In geometry Routh's theorem states the following: Let "ABC" be a triangle with area " [ABC] ". Let "F", "D" and "E" be points in the sides "AB", "BC" and "AC" such that the ratios "AF/BF", "BD/CD" and "CE/AE" are r, s and "t" respectively. Let "I", "G" and "H" be the intersection points of "AD" and "CF", "AD" and "BE", and "BE" and "CF" respectively. Then the area of triangle "GHI" is

: [GHI] =frac{(rst-1)^2}{(st+s+1)(rt+t+1)(rs+r+1)} [ABC] .

This theorem was given by Edward John Routh on page 82 of his "Treatise on Analytical Statics with Numerous Examples" in 1896. The particular case r = s = t = 2 has become popularized as the one-seventh area triangle.

References

* Murray S. Klamkin and A. Liu, Three more proofs of Routh's theorem, "Crux Mathematicorum" 7 (1981) 199–203
*H. S. M. Coxeter, "Introduction to Geometry", 2nd edition, Wiley, New York, 1969
*J. S. Kline and D. Velleman, Yet another proof of Routh's theorem, "Crux Mathematicorum" 21 (1995) 37–40

External links

* " [http://demonstrations.wolfram.com/RouthsTheorem/ Routh's Theorem] ," Jay Warendorff, The Wolfram Demonstrations Project.
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