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[parent] incircle radius determined by Pythagorean triple (Feature)

If the sides of a right triangle are integers, then so is the radius of the incircle of this triangle.

Proof. The sides of such a right triangle may be expressed by the integer parametres $m,\,n$ with $m > n > 0$ as

$\displaystyle a \;=\; 2mn, \quad b \;=\; m^2\!-\!n^2, \quad c \;=\; m^2\!+\!n^2;$ (1)

the radius of the incircle is
$\displaystyle r \;=\; \frac{2A}{a\!+\!b\!+\!c},$ (2)

where $A$ is the area of the triangle. Using (1) and (2) we obtain $$r \;=\; \frac{2\cdot2mn\cdot(m^2\!-\!n^2)/2}{2mn\!+\!(m^2\!-\!n^2)\!+\!(m^2\!+\!n^2)} \;=\; \frac{2mn(m\!+\!n)(m\!-\!n)}{2m(m\!+\!n)} \;=\; (m\!-\!n)n,$$ which is a positive integer.

Remark. The corresponding radius of the circumcircle need not to be integer, since by Thales' theorem, the radius is always half of the hypotenuse which may be odd (e.g. 5).



"incircle radius determined by Pythagorean triple" is owned by pahio.
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See Also: triangle, Pythagorean triplet, difference of squares, first primitive Pythagorean triplets, $x^4-y^4=z^2$ has no solutions in positive integers

Other names:  incircle radius of right triangle

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Cross-references: odd, hypotenuse, Thales theorem, circumcircle, positive, area, parametres, proof, triangle, incircle, radius, integers, right triangle, sides

This is version 9 of incircle radius determined by Pythagorean triple, born on 2008-01-27, modified 2009-08-20.
Object id is 10216, canonical name is IncircleRadiusDeterminedByPythagoreanTriple.
Accessed 1552 times total.

Classification:
AMS MSC: 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)
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