Menelaus' Theorem

For Triangles in the Plane,

$$AD\cdot BE\cdot CF=BD\cdot CE\cdot AF.$$ (1)

For Spherical Triangles,

$$\sin AD\cdot \sin BE\cdot \sin CF=\sin BD\cdot \sin CE\cdot \sin AF.$$ (2)

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

$$\prod_{i=1}^n \left[{V_iW_i\over W_iV_{i+1}}\right]=(-1)^n.$$ (3)

Here, $AB\vert\vert CD$ and

$$\left[{AB\over CD}\right]$$ (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

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. 66-67, 1967.

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

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