Maximising a Ratio

Consider a Phytagorean right-angled triangle with lengths a, b, and c (a < b < c). Its Perimeter: P=a+b+c Its Area: A=a*b/2 Find the (a,b,c) triple with maximum A/P ratio if P<=10^17. Answer format: a,b,c Example: 280,294,406 if P<=10^3 A=41160 P=980 A/P=42 [My timing: 40s]