We construct Fuchsian groups [Gamma] of signature (0 : 2, ... ,2 ;1;0) so that the set of hyperbolic fixed points of [Gamma] will contain a given finite collection of elements in the boundary of the hyperbolic plane. We use this to establish that there are infinitely many non-commensurable non-cocompact Fuchsian groups [Delta] of finite covolume sitting in PSL₂(Q) so that the set of hyperbolic fixed points of [Delta] will contain a given finite collection of rational boundary points of the hyperbolic plane. We also give a parameterization of Fuchsian groups of signature (0:2,2,2;1;0) and investigate when particular hyperbolic elements have rational fixed points. Moreover, we include a detailed list of the group elements and their killer intervals for the known pseudomodular groups that Long and Reid found; in addition, the list contains a new list of killer intervals for a pseudomodular group not found by Long and Reid. / text