Ceva's Theorem

Ceva's theorem states that, 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 Ceva (1678).

Let be an arbitrary polygon, 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).

Kiragu (2025, p. 26) gives three identities for nonconcurrent cevians. Let , , and lie strictly between and , and , and and , respectively, in a triangle . Suppose , , and are not concurrent, and put , , and . Also let , , and . Then

			(4)
			(5)
			(6)

where and segment symbols denote ordinary positive lengths. When the cevians are concurrent, the first identity remains valid with and its right-hand side is 1, recovering Ceva's theorem. For a short proof, both sides of the first identity equal

(7)

by the law of sines. Multiplying the three relations from Menelaus' theorem for the transversals , , and of triangle gives the second identity. Applying the first identity to triangle shows that , whose reciprocal gives the third identity.