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}}}$${\displaystyle =x}$, ${\displaystyle {\tfrac {AE}{CE}}}$${\displaystyle =y}$, and ${\displaystyle {\tfrac {BF}{AF}}}$${\displaystyle =z}$, the signed area of the triangle formed by the cevians ${\displaystyle AD}$, ${\displaystyle BE}$, and ${\displaystyle CF}$ is the area of triangle ${\displaystyle ABC}$ times

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