Find a triangle such that :
$area=S\in\mathbb{N^{*}}$ and the sides $a,b,c$ prime numbers
I need find this triangle not by imagine I need by a prof or something
I try $2,3,5$ , $3,5,11$ , $13,11,7$ but I tried
I would like a explain to find this sides
I have already see your hints and ideas
One possibility is that the area is $0$, such as your $2,3,5$ attempt. In fact, $2$ together with any twin prime works.
If we don't want the area to be $0$, then I think the most helpful tool we have at our disposal is Heron's formula for the area: $$ S=\sqrt{s(s-a)(s-b)(s-c)}\\ s=\frac{a+b+c}2 $$ We see that if $s$ is not an integer (and thus is half of some odd integer), then there is no hope for the area to be an integer either (as the radicand necessarily becomes an odd number divided by $16$). And since most primes are odd, that means one of the sides must be $2$. And since we don't want area $0$, we must have an isosceles triangle. Say $a=2$ and $b=c$.
So, with this, we get $s=b+1$, and Heron's formula becomes $$ S=\sqrt{(b+1)(b-1)\cdot1^2}\\ =\sqrt{b^2-1} $$ For this to be an integer, we need $b=1$, which 1) isn't a prime, and 2) gives us area $0$. So there are no positive integer area triangles with prime sides.