For triangles in the plane,

For spherical triangles,

This can be generalized to -gons , where a transversal cuts the side in for , ..., , by

Here, and

is the ratio of the lengths and with a plus or minus sign depending if these segments have the same or opposite directions (Grünbaum and Shepard 1995). The case 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 Resource. https://mathworld.wolfram.com/MenelausTheorem.html

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