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).

See also Ceva's Theorem ,

Hoehn's Theorem ,

Menelaus' Theorem ,

Schlicht
Function ,

Self-Transversality Theorem
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References Grünbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68 , 254-268, 1995. Krantz,
S. G. "Schlicht Functions." §12.1.1 in Handbook
of Complex Variables. Boston, MA: Birkhäuser, p. 149, 1999. Referenced
on Wolfram|Alpha Area Principle
Cite this as:
Weisstein, Eric W. "Area Principle." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/AreaPrinciple.html

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