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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, which gives the area of the black triangle in terms of the areas of the other three triangles.


	Contributed by: S. M. Blinder With corrections contributed by Liam Bauress and Oscar Chavez

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

. By the law 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 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

Reference: "The Eutrigon Theorem - a new* twin to the theorem of Pythagoras" on this website.

Contributed by: S. M. Blinder

With corrections contributed by Liam Bauress and Oscar Chavez

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