## Ceva's Theorem

Given a Triangle with 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)

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 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).

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 122, 1987.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1967.

Grünbaum, B. and Shepard, G. C. Ceva, Menelaus, and the Area Principle.'' Math. Mag. 68, 254-268, 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 145-151, 1929.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., p. xx, 1995.