Ceva — Ceva's theorem · Principia

Dependencies: Triangle area = ½ × base × height ( $\frac{1}{2}\,\text{base}\times\text{height}$ ), Ruler axiom.

Statement

Let $\triangle ABC$ have three cevians (segments from a vertex to a point on the opposite side)

AD\ (D \in \overline{BC}), \quad BE\ (E \in \overline{CA}), \quad CF\ (F \in \overline{AB}).

The three cevians are concurrent (i.e. they meet at a single point $P$ ) if and only if

\frac{|BD|}{|DC|} \cdot \frac{|CE|}{|EA|} \cdot \frac{|AF|}{|FB|} \;=\; 1.

$Ceva: three cevians AD, BE, CF through a point P inside \triangle ABC, with product of three ratios =1$