## Pythagorean triangle-perimeter a square and diameter of the inscribed circle a cube

Find a Pythagorean triangle with perimeter   $p$   a square and diameter of the inscribed circle a cube

Let the sides be

$(m^2 - n^2) \,x$,     $2 \, m \, n \, x$,     $(m^2 + n^2) \,x$

$p \; = \; 2 \, m \, x \, (m + n) \; = \; r^2$

Then     $r \; = \; n \, x \, (m - n)$

the diameter   $d$   is to be a cube,   say   $r^3 \,/ \,s^3$

$d \; = \; 2 \, n \, x \, (m - n) \; = \; r^3 \,/ \,s^3$

From the two values of   $x$

$x \; = \; r^2 \,/ \,(2 \, m \, (m+n))$
$x \; = \; r^3 \,/ \,(2 \, n \, s^3 \, (m-n))$

we get   $r$   in terms of   $m, n, s$
$r^2 \,/ \,(2 \, m \, (m+n)) \; = \; r^3 \,/ \,(2 \, n \, s^3 \, (m-n))$

$r \; = \; (m \, s^3 \, (n-m)) \,/ \,(n \, (m+n))$

## About benvitalis

math grad - Interest: Number theory
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