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Menelaus' Theorem

MenelausTheorem

For triangles in the plane,

AD·BE·CF=BD·CE·AF.
(1)

For spherical triangles,

sinAD·sinBE·sinCF=sinBD·sinCE·sinAF.
(2)

This can be generalized to n-gons P=[V_1,...,V_n], where a transversal cuts the side V_iV_(i+1) in W_i for i=1, ..., n, by

product_(i=1)^n[(V_iW_i)/(W_iV_(i+1))]=(-1)^n.
(3)

Here, AB∥CD and

[(AB)/(CD)]
(4)

is the ratio of the lengths [A,B] and [C,D] with a plus or minus sign depending if these segments have the same or opposite directions (Grünbaum and Shepard 1995). The case n=3 is Pasch's axiom.

See also

Ceva's Theorem, Hoehn's Theorem, Pasch's Axiom

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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. "Menelaus's Theorem." §3.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 66-67, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 42-44, 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. "The Theorem of Menelaus." Ch. 13 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 147-154, 1995.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., p. xxi, 1995.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 150, 1991.

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Menelaus' Theorem

Cite this as:

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

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