The Compleat Ceva

Extract

1. Ceva’s Theorem (Figure 1) is a familiar result in elementary geometry. It states that the product of three ratios of line segments on the sides of a triangle takes the fixed value 1. An n-gonal form of the theorem is also known, and this is just one of a large class of results concerning cyclic products of ratios of line segments on the sides or diagonals of an n-gon. In particular, suppose M is a fixed point (the Ceva point) and P = [V₀, … , V_n-1] is an n-gon. Consider the lines MV_i, joining the Ceva point to each of the vertices of P. These lines will cut the sides and diagonals of P in certain ratios, and each cyclic product of such ratios either takes a constant value or is related to another such cyclic product in a simple manner. The purpose of this note is to determine all such relations.