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Puzzle: In the following figure, what is the area of the green square?
Solution: First, observe that the largest right triangle (colored blue in the following diagram) measures r by 1 by sqrt(r2+1), in order of their side lengths.
Next, observe that the slightly smaller right triangle (colored blue near the bottom of the next illustration) is similar to the first, and so it measures r/sqrt(r2+1) by 1/sqrt(r2+1) by 1, in order of increasing side lengths.
The area of the smaller blue triangle is half the product of the legs, (1/2) r/(r2+1). Four such triangles are subtracted from 1 to get the area of the small square: 1-2r/(r2+1).
Mathworld: Square, which contains this puzzle
The webmaster and author of this Math Help site is Graeme McRae.