Menelaus's theorem

Dependencies: basic proportionality theorem (Basic proportionality (intercept theorem)), uniqueness of parallels (Through a point off a line, exactly one parallel exists (Playfair)).

Statement

Let $\triangle ABC$ be given, and let a transversal $\ell$ meet the lines containing the three sides at points $D$, $E$, $F$ — concretely:

with the convention that at least one of $D$, $E$, $F$ lies on a side's extension (otherwise $\ell$ cannot both pass through the triangle and meet the interiors of all three sides). Then the product of the three division ratios equals $1$ (unsigned):

Using signed ratios (taking signs along a fixed direction), the product equals $-1$ — exactly the "dual" of the $+1$ given by Ceva's theorem for the concurrent case.