Area Principle

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

The geometric area principle states that

This can also be written in the form

where

is the ratio of the lengths [A,B] and [C,D] for AB∥CD with a plus or minus sign depending on if these segments have the same or opposite directions, and

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 f is a schlicht function and if

then

(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.

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Area Principle

Cite this as:

Weisstein, Eric W. "Area Principle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AreaPrinciple.html

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