When the circle's radius is an integer
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Each Pythagorean triple (a, b, c) can be associated with a right triangle T. The radius of its incircle is also an integer. For example, the triple (3, 4, 5) corresponds to the right triangle with side lengths 3, 4 and 5, whose incircle has radius r = 1. The desired diameter r is in fact given by the formula 2r = a + b – c, where c is the hypotenuse, a is odd and gcd(a, b) = 1.
This elegant result seems to be little known. It nevertheless appears explicitly in an exercise proposed by Dominique Roux.
It follows very neatly from the arithmetic characterization of Pythagorean triples (see opposite). You can also have fun deriving the result by purely geometric means, using a clever dissection of the figure that reveals several similar right triangles!
What's new in mathematics? Pythagoras!
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