Collinear CM-points

André's celebrated theorem of 1998 implies that each complex straight line Ax+By+C=0 (apart from obvious exceptions) contains at most finitely many points (j(τ),j(τ')), where τ,τ'∈H are algebraic of degree 2 We show that there are only a finite number of such lines which contain more than two such points. As there is a line through any two complex points, this is the best possible result.