Let three quadratic equations ax2−2bx+c=0,bx2−2cx+a=0 and cx2−2ax+b=0, all have only positive roots. Then which of these are always true?

For each equation, required conditions(i) Discriminant≥0⇒b2≥ac,c2≥ab,a2≥bc(ii) f(0)>0 ⇒c,a,b>0(iii) Abscissa of vertex > 0 ⇒ba>0,cb>0,ac>0.Now, b2≥ac,c2≥ab⇒b2c2≥a2bc⇒bc≥a2But a2≥bc. So, a2=bc.Similarly, b2 = ac and c2 = ab.Therefore, a = b = c.