## Pythagorean triangle-perimeter + square of any side is a square

Find a Pythagorean triangle the sum whose perimeter and square of any side is a square

Let the sides be   $a \,x, \; b \,x, \; c \,x$,   where   $a^2 + b^2 = c^2$

Then

$x^2 \; + \; p \,x$
$x^2 \; + \; q \,x$
$x^2 \; + \; r \,x$

where    $p = s/a^2$,    $q = s/b^2$,    $r = s/c^2$,
$s = a + b + c$

math grad - Interest: Number theory
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### 2 Responses to Pythagorean triangle-perimeter + square of any side is a square

1. Paul says:

Here are a few, all of them are with a^2.

{16,63,65} = 144 + 16^2 = 20^2
{64,510,514} = 1088 + 64^2 = 72^2
{144,1725,1731} = 3600 + 144^2 = 156^2
{256,4092,4100} = 8448 + 256^2 = 272^2
{400,7995,8005} = 16400 + 400^2 = 420^2
{576,13818,13830} = 28224 + 576^2 = 600^2
{784,21945,21959} = 44688 + 784^2 = 812^2
{1024,32760,32776} = 66560 + 1024^2 = 1056^2
{1296,46647,46665} = 94608 + 1296^2 = 1332^2
{1936,85173,85195} = 172304 + 1936^2 = 1980^2
{2304,110580,110604} = 223488 + 2304^2 = 2352^2
{2704,140595,140621} = 283920 + 2704^2 = 2756^2
{3136,175602,175630} = 354368 + 3136^2 = 3192^2
{3600,215985,216015} = 435600 + 3600^2 = 3660^2
{4096,262128,262160} = 528384 + 4096^2 = 4160^2
{4624,314415,314449} = 633488 + 4624^2 = 4692^2
{5184,373230,373266} = 751680 + 5184^2 = 5256^2
{5776,438957,438995} = 883728 + 5776^2 = 5852^2
{6400,511980,512020} = 1030400 + 6400^2 = 6480^2
{7056,592683,592725} = 1192464 + 7056^2 = 7140^2
{7744,681450,681494} = 1370688 + 7744^2 = 7832^2
{8464,778665,778711} = 1565840 + 8464^2 = 8556^2
{9216,884712,884760} = 1778688 + 9216^2 = 9312^2
{10000,999975,1000025} = 2010000 + 10000^2 = 10100^2

Paul.