How can I prove that the points A,B,C representing the complex numbers
a,b,c are on a circle that goes through O if 2/a=1/b+1/c ?

Hint:  What would be the center of such a circle?  its radius?

Any three points in the plane which are not in a straight line
determine a unique circle passing through them.  Actually straight
lines and circles in the complex plane are closely related, and
knowing a few facts about this relationship would perhaps provide a
quick shortcut:

  1/a = (1/b + 1/c)/2

regards, mathtalk-ga