Under twice continuous differentiability assumptions on the generator functions \(f\) and \(g\), the authors determine all two variable quasi-arithmetic means \(M^{[f]}\), \(M^{[g]}\) and all real numbers \(\lambda\) and \(\mu\) such that \[ \lambda M^{[f]}+\mu M^{[g]}=A, \] where \(A\) denotes the two variable arithmetic mean.