To be sure, this exists.

If you want to make the areas of △ABM and △ABC equal and take AB as the base, then the distance from point M to AB is equal to the distance from point C to AB, that is, the height.

After point C, pan to the left to M, then connect CM to make CM∨AB see the degree of ∠AMB at this time. The closer point M is to point C, the greater the degree, and vice versa. In this way, the translation of point M to the left can definitely reach 60.

It can be concluded that there are four such points, one on the left of point C, one on the right, and the other two below the AB side.

If you don't understand what I said, contact me in time.

I hope I can help you.