The ``Area principle'' states that

(1) 
This can also be written in the form

(2) 
where

(3) 
is the ratio of the lengths and for with a Plus or Minus Sign
depending on if these segments have the same or opposite directions, and

(4) 
is the Ratio of signed Areas of the Triangles. Grünbaum and Shepard show that
Ceva's Theorem, Hoehn's Theorem, and Menelaus' Theorem are the consequences of this result.
See also Ceva's Theorem, Hoehn's Theorem, Menelaus' Theorem, SelfTransversality Theorem
References
Grünbaum, B. and Shepard, G. C. ``Ceva, Menelaus, and the Area Principle.'' Math. Mag. 68, 254268,
1995.
© 19969 Eric W. Weisstein
19990525