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Ceva's Theorem


CevasTheorem

Given a triangle with polygon vertices A, B, and C and points along the sides D, E, and F, a necessary and sufficient condition for the cevians AD, BE, and CF to be concurrent (intersect in a single point) is that

 BD·CE·AF=DC·EA·FB.
(1)

This theorem was first published by Giovanni Ceva 1678.

Let P=[V_1,...,V_n] be an arbitrary n-gon, C a given point, and k a positive integer such that 1<=k<=n/2. For i=1, ..., n, let W_i be the intersection of the lines CV_i and V_(i-k)V_(i+k), then

 product_(i=1)^n[(V_(i-k)W_i)/(W_iV_(i+k))]=1.
(2)

Here, AB∥CD and

 [(AB)/(CD)]
(3)

is the ratio of the lengths [A,B] and [C,D] 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.

Cite this as:

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

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