Solving a Geometric Problem with Friends

[henry1964]

A couple friends worked on this problem (for a week now...)

Trying to show that a conformal bijective map that sends vertices of one rectangle to vertices of another rectangle on the complex plane has to be linear.
I would appreciate any help, Thank you.

 

[Hurkyl]

Well, the map is completely determined by its action on three points, right? Just check every possible case by hand. One or more should be "obvious", geometrically speaking.