Difficult problem - Analytical geometry

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This is an interesting problem in my opinion. I tried to solve it but without considerable success. A colleague of mine showed it few years ago. He said that the problem may have a practical application - for example for the spherical mirrors on the roads. I like this problem because it is combination between beauty of the math and its practical use.

It is given a circumference k with center O(xo, yo) and radius R, and a segment AB it is defined by the points - A(x1, y1) and B(x2,y2). Find the locus of all the points M lying on k for which the line OM is an angle bisector of the angle AMB.