We have $b^2=a(a+c)$ and $a^2+b^2<c^2$ $a|b^2$ so $a=xy^2,b=xyz$ for some integers $x,y,z$ $c=x(z^2-y^2)$ and we have $c<a+b \to z^2-y^2<y^2+yz \to z<2y$ and $c^2>a^2+b^2 \to (z^2-y^2)^2>y^4+y^2z^2 \to z^2>3y^2$ so $(2y-1)^2 \geq z^2>3y^2 \to y \geq 4 \to z\geq 7$ And $P=a+b+c=x(z^2+yz) \geq 77$ (for $(a,b,c)=(16,28,33)$)