Menelaus’ theorem

Menelaus' theorem is a geometric theorem that relates the points where a straight line intersects the sides of a triangle. If a transversal line cuts the sides of a triangle ABC (or their extensions) at points P, Q, and R on sides BC, AB, and AC respectively, then the product of the lengths of the segments Aq × Bp × Cr is equal to the product of the lengths Bq × Cp × Ar.1

This theorem was established by the Greek mathematician Menelaus of Alexandria in the 1st century AD, as part of his work on spherical triangles. While Menelaus' theorem is fundamental in plane geometry, it also has a form applicable to spherical triangles, which was crucial for the development of spherical trigonometry and astronomy. It is considered dual to Ceva's theorem, which deals with concurrency of lines within a triangle.23