## Primitive isosceles Heronian triangle with an even side of the form 4n + 2

Any number of the form   $4 \,n + 2$   may be used as the even side of a primitive isosceles Heronian triangle by using sides

$a \; = \; 2 \, n^2 \; + \; 2 \, n \; + \; 1$
$b \; = \; 2 \, n^2 \; + \; 2 \, n \; + \; 1$
$c \; = \; 4 \, n \; + \; 2$

Let  $P$   be the perimeter:

$P \; = \; 2 \,(2 \, n^2 + 2 \, n + 1) + 4 \, n + 2 \; = \; 4 \,(n + 1)^2$

If   $A$   is the area of a triangle whose sides have lengths   $a, \; b$,   and   $c$   then

$4 \, A \; = \; \sqrt { \,(a+b+c) \,(a+b-c) \,(a-b+c) \,(-a+b+c) \,}$

Let’s compute the area of the triangle

$a + b + c \; = \; 4 \, (n + 1)^2$
$a + b - c \; = \; 4 \, n^2$
$a - b + c \; = \; 2 \,(2 \, n + 1)$
$-a + b + c \; = \; 2 \,(2 \, n + 1)$

$4 \, A \; = \; \sqrt { \,4 \, (n+1)^2 \: (4 \, n^2) \: 4(2 \, n + 1)^2 \,}$
$4 \, A \; = \; 8 \, n \,(n + 1) \,(2 \, n + 1)$
$A \; = \; 2 \, n \,(n + 1) \,(2 \, n + 1)$

we find that it’s an integer.

Here are the first few primitives: