Area Principle

There are at least two results known as "the area principle."

The geometric 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 (1995) show that Ceva's theorem, Hoehn's theorem, and Menelaus' theorem are the consequences of this result.

The area principle of complex analysis states that if is a schlicht function and if

(5)

then

(6)

(Krantz 1999, p. 150).