If a, b, c  are three real numbers such that a+b+c=0 (at least one a, b, c is not equal to zero) and az1+bz2+cz3=0, then z1,z2,z3 :

az1+bz2+cz3=0, a+b+c=0.  at least one of a, b, c≠0 and a, b,c∈R . Let a≠0 Then z1=bz2+cz3b+c                      [∵   −a=b+c] Thus, z1 divides the line segment joining z2  and z3 in the ratioc  :  b and therefore z1,z2,z3 are collinear