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.

Remarks

The signed ratio $\frac{AF}{FB}$ is positive if $F$ is between $A$ and $B$, and negative otherwise. The theorem uses only ratios of segments on one line. So it is true in each affine plane over each field. For the three medians, each ratio is $1$. The converse then shows again that the medians meet in one point.

History

Giovanni Ceva published the theorem in 1678, in his work De lineis rectis. Yusuf al-Mu'taman ibn Hud, an 11th century king of Zaragoza, had proved it much earlier.

Explanations

L1 human

Draw three lines from the corners of a triangle through one common inside point. Each line cuts the opposite side into two parts. Multiply the three ratios of the parts, going round the triangle in one direction: the product is always $1$.