Ceva's theorem

Statement

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$. For collinear points $X$, $Y$, $Z$ with $Z \neq Y$, let the signed ratio $\frac{XZ}{ZY}$ be the real number $\lambda$ with $Z - X = \lambda\,(Y - Z)$. Let $O$ be a point that is not on the lines $BC$, $CA$ and $AB$. Suppose that the lines $AO$, $BO$ and $CO$ meet the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$. Then $$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.$$ Conversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $1$. Then the lines $AD$, $BE$ and $CF$ meet in one point or are all parallel.