Be a Hero

Ok, let me just say, you could guess and check this problem way quicker than if you did it the legit way. But that's weak, so we're not gonna do that.

Heron's formula tells us that $P^2=S(S-A)(S-B)(S-C)$ . Substituting $S=\frac{a+b+c}{2}$ (S=semi perimeter)

$P^2=\frac{a+b+c}{2}(\frac{a+b+c}{2}-\frac{2A}{2})(\frac{a+b+c}{2}-\frac{2B}{2})(\frac{a+b+c}{2}-\frac{2C}{2})$ . Simplifying some more.

$P^2=\frac{a+b+c}{2}(\frac{-a+b+c}{2})(\frac{a-b+c}{2})(\frac{a+b-c}{2})$ Some more

$16P^2=(A+B+C)(-A+B+C)(A-B+C)(A+B-C)$ . Notice how similar this equation is to the one we started with. The only difference is that the original equation has -1 factored into two of the trinomials.

$16P^2=(-1)(A+B+C)(-1)(-A+B+C)(A-B+C)(A+B-C)$ distributing yields

$16P^2=(-A-B-C)(A-B-C)(A-B+C)(A+B-C)$ .

Thus K=16

If my solution is incomplete or if you have any questions, feel free to ask and comment.

We all know Heron's formula of area of triangle with 3 sides.

Solution