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'A Brief Introduction to the Argand Diagram' printed from http://nrich.maths.org/
Pick a point for $z_1$ such as $2+i$. Move $z_2$ around, and keep a record of the points where $z_3$ is on the x-axis. What do you notice about the coordinates of $z_2$ and $z_3$?
Repeat for some other points.