I just figured out how to find the midpoint between two points using just a compass and no straight edge. A similar approach can be found in this question:
Constructing the midpoint of a segment by compass
Now what I am wondering is, can I find the circumcenter of a scalene triangle using just a compass and no straight edge?
I am thinking of drawing a perpendicular at C (in the diagram in the link of the answer presented by commenter and has nine likes) and choosing more points as my centres on that line to create a circle with points A and M on it. So in essence, I would be transforming a line (the perpendicular) into a circle, which I can draw using a compass. Yet, I do not really know if and how it helps.
Best of luck!