We study the problem of locating a new homogeneous facility under a prelocated facility. Here, a group of agents are located on the real line, each of whom has her location as private information and her cost is the (expected) distance from her location to the nearest facility. Our goal is to design mechanisms which can approximately minimize the maximum cost and the social cost while eliciting agents’ private information truthfully (i.e. strategy-proof). Based on the real-life scenarios, we consider the problem in two settings: the general setting where each agent can be located at both sides of the prelocated facility, and the special setting where all the agents are located at the same side of the prelocated facility. In the general setting, we design the best possible deterministic strategy-proof mechanism with 2-approximation and provide a lower bound of \(1.5-\epsilon (\epsilon >0)\) for any randomized strategy-proof mechanism under the maximum cost objective. For the social cost, we obtain an upper bound of n for deterministic strategy-proof mechanisms, and lower bounds of 1.5 and 1.0425 for any deterministic strategy-proof mechanism and any randomized strategy-proof mechanism, respectively. In the special setting, we further provide a randomized strategy-proof 5/3-approximate mechanism for the maximum cost and a deterministic strategy-proof \((n-1)\) -approximate mechanism for the social cost.