If $z_1=1+2i,z_2=2+3i,z_3=3+4i,$ then $z_1,z_2,z_3$ represent the vertices of a/an 

$z_1=1+2i,z_2=2+3i,z_3=3+4i$

$|z_1-z_2|=|-1-i|=\sqrt{2}$

and  $|z_2-z_3|=-1-i=\sqrt{2}$

and $|z_1-z_3|=|-2-2i|=2\sqrt{2}$.

Hence vertices are collinear.