If z1,z2,z3 are three distinct complex numbers and  a,b,c are there positive real numbers such that a|z2−z3|=bz3-z1)=cz1−z2 then a2z2-z3+b2(z3−z1)+c2(z1−z2)=

Let a|z2−z3|=b|z3−z1|=c|z1−z2|=λ  (say) ⇒a=λ|z2−z3|,b=λ|z3−z1|,c=λ|z1−z2| ⇒a2=λ2|z2−z3|2=λ2(z2−z3)(z¯2−z¯3) ∴a2(z2−z3)=λ2(z¯2−z¯3) Similarly, b2(z3−z1)=λ(z¯3−z¯1) and c2|z1−z2|=λ2(z¯1−z¯2) ∴a2z2−z3+b2z3−z1+c2z1−z2=0 .