Ceva's Theorem

Given a triangle with polygon vertices , , and and points along the sides , , and , a necessary and sufficient condition for the cevians , , and to be concurrent (intersect in a single point) is that

(1)

This theorem was first published by Giovanni Ceva 1678.

Let be an arbitrary -gon, a given point, and a positive integer such that . For , ..., , let be the intersection of the lines and , then

(2)

Here, and

(3)

is the ratio of the lengths and with a plus or minus sign depending on whether these segments have the same or opposite directions (Grünbaum and Shepard 1995).

Another form of the theorem is that three concurrent lines from the polygon vertices of a triangle divide the opposite sides in such fashion that the product of three nonadjacent segments equals the product of the other three (Johnson 1929, p. 147).