## Root-Mean-Square| Primitive Triangle(a,b,c) such that a^2 + c^2 = 2b^2

Root-Mean-Square
http://mathworld.wolfram.com/Root-Mean-Square.html

parametric representation of all primitive solutions   $(a,b,c)$   of the Diophantine equation:

$a^2 \; + \; c^2 \; = \; 2 \,b^2$,     $a > b > c$

$a \; = \; (u^2 + 2 \,u \,v - v^2) \, t$
$b \; = \; (u^2 + v^2) \, t$
$c \; = \; \pm \, (u^2 - 2 \,u \,v - v^2) \, t$

where   $u, v$   are relatively prime, with   $u > v$   and   $t$   an integer

$a^2 \; + \; c^2$
$= \; ((u^2 + 2 \,u \,v - v^2) \, t)^2 \; + \; ((u^2 - 2 \,u \,v - v^2) \, t)^2$
$= \; 2 \, u^4 \, t^2 \; + \; 4 \, u^2 \, v^2 \, t^2 \; + \; 2 \, v^4 \, t^2$
$= \; 2 \,(u^4 \, t^2 \; + \; 2 \, u^2 \, v^2 \, t^2 \; + \; v^4 \, t^2)$
$= \; 2 \,(t^2 \, (u^2 \; + \; v^2)^2)$
$= \; 2 \, b^2$

Perimeter where   $t = 1$ :

$a \; + b \; + \; c$
$= (u^2 + 2 \, u \, v - v^2) \; + \; (u^2 + v^2) \; + \; (u^2 - 2 \, u \, v - v^2)$
$= 3 \, u^2 \; - \; v^2$

OR

$= (u^2 + 2 \, u \, v - v^2) \; + \; (u^2 + v^2) \; - \; (u^2 - 2 \, u \, v - v^2)$
$= u^2 \; + \; 4 \, u \, v \; + \; v^2$