## Isosceles Triangle whose Ratio Area/Perimeter is a prime number

For example,

Can you find another isosceles triangle whose sides are integers and

Area / Perimeter = p

where p is a prime number?

Note that

$Area \; = \; b/4 \, \sqrt{4 \, a^2 - b^2}$
$Perimeter \; = \; (2 \, a + b)$

and

$b/4 \sqrt{4 \, a^2 - b^2} \; = \; p \, (2 \, a + b)$

$b \, \sqrt{4 \, a^2 - b^2} \; = \; 4 \, p \, (2 \, a + b)$

squaring both sides,

$b^2 \, (4 \, a^2 - b^2) \; = \; 16 \, p^2 \, (2 \, a + b)^2$

$b^2 \, (2 \, a - b) \,(2 \, a + b) \; = \; 16 \, p^2 \, (2 \, a + b)^2$

$b^2 \, (2a - b) \; = \; 16 \, p^2 \, (2 \, a + b)$

$b^2 \, 2 \, a \; - \; b^3 \; = \; 32 \, p^2 \, a \; + \; 16 \, p^2 \, b$

$b^2 \, 2 \, a \; - \; 32 \, p^2 \, a \; = \; 16 \, p^2 \, b \; + \; b^3$

$a \, (2 \, b^2 - 32 \, p^2) \; = \; (16 \, p^2 \, b + b^3)$

$a \; = \; (16 \, p^2 \, b + b^3) / (2 \, b^2 - 32 \, p^2)$