Question

Does the center of mass of a system of two particles lie on the lines joining the particles?

Answer 1

Yes,

If the ends of joining line, crosses both center of mass of particle.

e.g. two spheres of equal volume and mass join with a line,

line is joins in such a way that it crosses center of both spheres.

Then the center of mass of a system of two particles lie on the lines joining the particles.