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

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

Here, and

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).

See also

Hoehn's Theorem, Menelaus' Theorem

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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. "Ceva's Theorem." §1.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1967.Durell, C. V. A Course of Plane Geometry for Advanced Students, Part I. London: Macmillan, p. 54, 1909.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 40-41, 1928.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 81, 1930.Grünbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254-268, 1995.Honsberger, R. "Ceva's Theorem." §12.1 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 136-138, 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.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 28-29, 1991.

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Ceva's Theorem

Cite this as:

Weisstein, Eric W. "Ceva's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CevasTheorem.html

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