TOPICS
Search

Routh's Theorem

If the sides of a triangle are divided in the ratios lambda:1, mu:1, and nu:1, the cevians form a central triangle whose area is

A=((lambdamunu-1)^2)/((lambdamu+lambda+1)(munu+mu+1)(nulambda+nu+1))Delta,
(1)

where Delta is the area of the original triangle. for lambda=mu=nu=n,

A=((n-1)^2)/(n^2+n+1)Delta.
(2)

for n=1, 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

A^'=(lambdamunu+1)/((lambda+1)(mu+1)(nu+1))Delta.
(3)

Routh's theorem gives Ceva's theorem and Menelaus' theorem (lambdamunu=-1) as special cases.

See also

Ceva's Theorem, Cevian, Menelaus' Theorem

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Routh's Theorem

Cite this as:

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

Subject classifications