Let a=tan(A∕2),b=tan(B∕2),c=tan(C∕2) all positive, the constraint becomes c=(1−ab)∕(a+b) Which is equivalent to ab+bc+ca=1 And we know that, a2+b2+c2≥ab+bc+ca ⇒a2+b2+c2≥1 ⇒tan2(A∕2)+tan2(B∕2)+tan2(C∕2)≥1 Therefore, Answer is k≥1