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

$$A={(\lambda\mu\nu-1)^2\over (\lambda\mu+\lambda+1)(\mu\nu+\mu+1)(\nu\lambda+\nu+1)} \,\Delta,$$ (1)

where $\Delta$ is the Area of the original Triangle. For $\lambda=\mu=\nu\equiv n$,

$$A={(n-1)^2\over n^2+n+1}\Delta.$$ (2)

For $n=1$, 2, 3, ..., the areas are 0, 1/7, 4/13, 3/7, 16/31, 25/43, .... The Area of the Triangle formed by connecting the division points on each side is

$$A'={\lambda\mu\nu\over (\lambda+1)(\mu+1)(\nu+1)}\,\Delta.$$ (3)

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

See also Ceva's Theorem, Cevian, Menelaus' Theorem

References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 211-212, 1969.