The Eutrigon Theorem


	The black central triangle with one angle equal to 60° is called a eutrigon. The areas of equilateral triangles constructed on the three faces , and, obey the eutrigon theorem, giving the area of the black triangle in terms of the other three triangles.


	Contributed by: S. M. Blinder

Here is a quick proof: The area of an equilateral triangle of side

. By the laws of sines, the area of a eutrigon is

. The law of cosines gives

, because

. Multiplying by

gives the statement of the theorem.

Snapshot 1: when

, the figure reduces to four equal equilateral triangles. The validity of the theorem then becomes trivial

Snapshot 2: when

, the eutrigon becomes a 30-60-90 right triangle. By an analog of Pythagoras' theorem,

, implying that the black triangle is twice the area of the blue triangle

Snapshot 3: a degenerate case, with

, which collapses the eutrigon

For a proof and discussion of the eutrigon theorem, see W. Roberts' website.

Law of Cosines (Wolfram MathWorld)

Pythagorean Theorem (Wolfram MathWorld)

Triangle (Wolfram MathWorld)

"The Eutrigon Theorem" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/TheEutrigonTheorem/

Contributed by: S. M. Blinder

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