In this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal first eigenproblem, we describe an operator splitting method applied to the corresponding time-dependent problem. For solving higher eigenproblems, we describe an arclength continuation method applied to a particular perturbation of the original problem, which admits solution branches bifurcating from the trivial solution branch at eigenvalues of its linearization. We then solve the original eigenproblem by ‘jumping’ to a point on the unperturbed solution branch from a ‘nearby’ point on the corresponding continued perturbed branch, then normalizing the result. Finally, for comparison, we describe a particular implementation of Newton's method applied directly to the original constrained nonlinear eigenproblem.