In this paper, firstly, we calculate Cauchy-length formula for the one-parameter planar motion in generalized complex plane C-p which is generalization of the complex, dual and hyperbolic planes. Then, we give the length of the enveloping trajectories of lines C-p. In addition, we prove the Holditch theorem for the non-linear three points with the aid of the length of the enveloping trajectories in C-p. So, the Holditch theorem for the linear three points which is given by Erisir et al. in C-p is generalized for trajectories drawn by the non-linear three points in generalized complex plane C-p.