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# Routh's Theorem

If the sides of a triangle are divided in the ratios , , and , the cevians form a central triangle whose area is

 (1)

where is the area of the original triangle. for ,

 (2)

for , 2, 3, ..., the areas are 0, 1/7 (Steinhaus 1999, pp. 8-9), 4/13, 3/7, 16/31, 25/43, ... (OEIS A046162 and A046163).

The area of the triangle formed by connecting the division points on each side is

 (3)

Routh's theorem gives Ceva's theorem and Menelaus' theorem () as special cases.

Ceva's Theorem, Cevian, Menelaus' Theorem

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## References

Bottema, O. "On the Area of a Triangle in Barycentric Coordinates." Crux. Math. 8, 228-231, 1982.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 211-212, 1969.Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 27, 1970.Klamkin, M. S. and Liu, A. "Three More Proofs of Routh's Theorem." Crux Math. 7, 199-203, 1981.Mikusiński, J. G. "Sur quelques propriétés du triangle." Ann. Univ. M. Curie-Sklodowska Sect. A 1, 45-50, 1946.Sloane, N. J. A. Sequences A046162 and A046163 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

Routh's Theorem

## Cite this as:

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