Ceva & Menelaus

Concurrency and collinearity are products of ratios.

Ceva's theorem: cevians $AD$, $BE$, $CF$ pass through one point exactly when $\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB} = 1$. Menelaus' theorem: the same product equals $1$ when the three points are cut by one straight line instead. Two of geometry's central questions — do three lines meet, do three points align — reduce to checking a single multiplication.

The configurations are told apart by counting division points that fall outside their segments: an even number means cevians (Ceva), an odd number means a transversal (Menelaus). With signed ratios the counts become the signs $+1$ and $-1$ and the two theorems merge into one.

Mass points are the fast interface: put masses on the vertices so each cevian foot is a balance point, and every ratio in the figure — including how the concurrency point splits each cevian, which Ceva alone cannot see — becomes a quotient of masses.

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