If the equations b+cx+c+ay+a+bz=0, cx+ay+bz=0, ax+by+cz=0 have non zero solutions, then a relation among a,b,c is

The condition for a homogeneous system to have a non-trivial solution is Δ=0⇒b+cc+aa+bcababc=0      R1→R1+R3⇒a+b+cb+c+aa+b+ccababc=0⇒a+b+c111cababc=0⇒a+b+cac-b2-c2-ab+bc-a2=0⇒a+b+cab+bc+ca-a2-b2-c2=0⇒a+b+ca2+b2+c2-ab-bc-ca=0 ⇒a+b+c=0 or a=b=c