How do I find the coordinates of B and D?

That would appear to be a formula for one of the other vertices, which can be derived from the methods you have been told. Where did you get it?
My instructor used this, but I had no idea how it was derived.
Since you do know about vectors, have you learned anything about rotating vectors, or about perpendicular vectors?
Unfortunately, not yet.
 
Here's the kicker: I understood most of it, except the part he used the circle's equation? Why is that usage needed here?

A circle is the set of all points a given distance from some central point, in this case M. We wanted the points that are the same distance from M that A and C are, but lying along the line perpendicular to the line containing A and C.
 
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