I assume you're using just a compass and straightedge. This seems too simple, and maybe flawed, but here goes. Maybe someone with more construction skills can jump in.
Step 1. Construct Circle O with desired radius
Step 2. Locate point P on circle and draw OP and extend past the circle
Step 3. Construct a perpendicular at P and extend (Remember the tangent is perpendicular to the radius at the point of tangency)
Step 4. Construct a segment PR congruent to PO forming equilateral triangle POR (each angle is 60 deg)
Step 5. Construct OR and extend intersecting tangent line at X.
Triangle POX is a 30-60-90 right triangle.
Step 6. Duplicate the 30 deg angle at X and extend to intersect circle at Y, which is the other point of tangency from X
XP = XY and they are 60 deg apart.
Pardon my crude drawing.