## Heron triangles of the form (u^2 + 2w^2, u^2 + 4w^2, 2u^2 + 2w^2)

If   $x \; = \; u^2$   and   $y \; = \; 2 \, w^2$

$x + y$,   $x + 2 \, y$,   $2 \, x + y$   are the sides of a triangle with rational area   $= \; (x+y) \: \sqrt { \,(2 \, x \, y) \,}$

$a \; = \; x \; + \; y \; = \; u^2 \; + \; 2 \, w^2$
$b \; = \; x \; + \; 2 \, y \; = \; u^2 \; + \; 4 \, w^2$
$c \; = \; 2 \, x \; + \; y \; = \; 2 \, u^2 \; + \; 2 \, w^2$

$s \; = \; (a + b + c)/2$
$= \; (x + y + x + 2 \,y + 2 \,x + y)/2 \; = \; 2 \, (x+y)$

$s \; - \; a \; = \; 2 \, (x+y) \; - \; (x + y) \; = \; x \; + \; y$
$s \; - \; b \; = \; 2 \, (x+y) \; - \; (x + 2 \,y) \; = \; x$
$s \; - \; c \; = \; 2 \, (x+y) \; - \; (2 \,x + y) \; = \; y$

$\sqrt { \,(s \,(s - a) \,(s - b) \,(s - c)) \,}$
$= \; \sqrt { \,(2 \,(x+y) \,(x + y) \,(x) \,(y)) \,}$
$= \; (x+y) \: \sqrt { \,(2 \, x \, y) \,}$

$s \; = \; (a + b + c)/2$
$= \; (u^2 + 2 \, w^2 + u^2 + 4 \, w^2 + 2 \, u^2 + 2 \, w^2)/2$
$= \; 2 \, (u^2 + 2 \, w^2)$

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

$\sqrt { \,(s \,(s - a) \,(s - b) \,(s - c)) \,}$
$= \; \sqrt { \,(2 \,(u^2 + 2 \,w^2) \,(u^2 + 2 \,w^2) \,(u^2) \,(2 \,w^2)) \,}$
$= \; 2 \, u \, w \, (u^2 + 2 \, w^2)$