Routh's theorem

In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ${\displaystyle ABC}$ points ${\displaystyle D}$, ${\displaystyle E}$, and ${\displaystyle F}$ lie on segments ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, then writing ${\displaystyle {\tfrac {CD}{BD}}=x}$, ${\displaystyle {\tfrac {AE}{CE}}=y}$, and ${\displaystyle {\tfrac {BF}{AF}}=z}$, the signed area of the triangle formed by the cevians ${\displaystyle AD}$, ${\displaystyle BE}$, and ${\displaystyle CF}$ is

${\displaystyle S_{ABC}{\frac {(xyz-1)^{2}}{(xy+y+1)(yz+z+1)(zx+x+1)}},}$

where ${\displaystyle S_{ABC}}$ is the area of the triangle ${\displaystyle ABC}$.

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 ${\displaystyle x=y=z=2}$ has become popularized as the one-seventh area triangle. The ${\displaystyle x=y=z=1}$ case implies that the three medians are concurrent (through the centroid).