The roots of the given equation are real so, D≥0 This implies, 4(a+b+c)2−4×3λ(ab+bc+ca)≥0 (a+b+c)2−3λ(ab+bc+ca)≥0 λ≤‌‌
(a+b+c)2
3(ab+bc+ca)
For scalane triangle, |b−c|<a,|c−a|<b,|a−b|<c So (b−c)2+(c−a)2+(a−b)2<a2+b2+c2 a2+b2+c2<2(ab+bc+ca) a2+b2+c2+2(ab+bc+ca)<a2+b2+c2<4(ab+bc+ca) ‌‌