Routh's theorem

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Routh's theorem in geometry 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 an 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].

[edit] References

  • M. 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
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