Ceva theorem

A theorem on the relation between the lengths of certain lines intersecting a triangle. Let $A_1,B_1,C_1$ be three points lying, respectively, on the sides $BC$, $CA$ and $AB$ of a triangle $ABC$. For the lines $AA_1$, $BB_1$ and $CC_1$ to intersect in a single point or to be all parallel it is necessary and sufficient that

$$\frac{AC_1}{C_1B}\cdot\frac{BA_1}{A_1C}\cdot\frac{CB_1}{B_1A}=1.$$

Lines $AA_1$, $BB_1$ and $CC_1$ that meet in a single point and pass through the vertices of a triangle are called Ceva, or Cevian, lines. Ceva's theorem is metrically dual to the Menelaus theorem. It is named after G. Ceva [1].

Ceva's theorem can be generalized to the case of a polygon. Let a point $0$ be given in a planar polygon with an odd number of vertices $A_1\dots A_{2n-1}$, and suppose that the lines $0A_1,\dots,0A_{2n-1}$ intersect the sides of the polygon opposite to $A_1,\dots,A_{2n-1}$ respectively in points $a_n,\dots,a_{2n-1}$, $a_1,\dots,a_{n-1}$. In this case

$$\frac{A_1a_1}{a_1A_2}\cdot\frac{A_2a_2}{a_2A_3}\cdots\frac{A_{2n-2}a_{2n-2}}{a_{2n-2}A_{2n-1}}\cdot\frac{A_{2n-1}a_{2n-1}}{a_{2n-1}A_1}=1.$$

References

[1]	G. Ceva, "De lineis rectis se invicem secantibus statica constructio" , Milano (1678)
[a1]	M. Berger, "Geometry" , I , Springer (1987)