Menelaus's theorem

Menelaus's theorem characterizes when three points on the sides or extensions of a triangle are collinear: the product of their directed side ratios is −1.

Directed line segment: A line segment equipped with an orientation, allowing its length to carry a positive or negative sign. Its sign convention determines why Menelaus's product is −1 rather than +1.
Triangle: A polygon formed by three line segments joining three noncollinear points. Its three side lines provide the framework for the theorem's points and ratios.
Menelaus configuration: A triangle together with a line meeting its three side lines at collinear points. This is the geometric arrangement whose ratios satisfy the theorem.
Menelaus of Alexandria: An ancient Greek mathematician and astronomer associated with spherical geometry and a theorem on transversals. The theorem is traditionally attributed to this mathematician.
Directed ratio: A quotient of signed lengths measured along an oriented line. The theorem multiplies three such ratios, one for each side of the triangle.
Side of a triangle: A line segment joining two vertices of a triangle. Each ratio compares distances along one side line, including its extensions.
Triangle side extension: The continuation of a triangle's side beyond one of its vertices. A transversal often meets an extended side, producing a negative directed ratio.
Sphaerica: A work by Menelaus of Alexandria on spherical geometry, preserved through later transmission. It contains the earliest surviving account associated with Menelaus's theorem.
Ceva's theorem: A triangle theorem stating that three cevians are concurrent exactly when a product of side ratios equals 1. It is the concurrency counterpart to Menelaus's collinearity criterion.
Line segment: The part of a line bounded by two endpoints. Side ratios are formed from segments whose endpoints include vertices and intersection points.

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