In the papers [1–5] using synthetic methods and the facts from the conics theory in the Euclidean, pseudo-Euclidean, quasi-hyperbolic and hyperbolic planes, we have studied and proved numerous facts related to the cuspidal curves with emphasis on evolutes and Steiner’s deltoid. In this paper, the intention is to see what can be said about the curve of centers in some special pencil of conics and to show how, studying the center curves in the special pencil of conics a connection with Steiner’s deltoid curve has been found.