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Proposition: The blue area equals the sum of the two pink areas.
The diagram above shows an equilateral triangle inscribed in a rectangle in such a way that the two have a vertex in common. This subdivides the rectangle into four disjoint triangles. The original equilateral triangle is shown in white in the diagram; the other three are shown in color.
Proposition The area of the blue triangle equals the sum of the areas of the two pink triangles.
The trigonometric proof is quite straightforward. I don't know of a classical proof a la Euclid. (Well, actually I haven't tried much.) If you can think of a neat non-trigonometric proof, let me know. I will put it here with due credit.
This problem appeared as a conjecture
in an article
geometry.puzzles newsgroup on March 15, 1997.
This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 2, 2010.
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