This paper proposes a real-time local positioning system (RT-LPs) utilizing a mobile platform equipped with three anchor nodes placed in a right-angle triangle formation for a real-time locating system (RTLS). After deriving an analytic formula to calculate the target position utilizing the measured distances among anchor nodes and the target node, we find that four parameters have an effect on the position error of the target node. The spacing between anchors is a design parameter that must be large enough to the reduce position error. However, the distance from the anchor node to the target node is an operation parameter that must be small enough to reduce the position error. Additionally, the measured ranges among the anchors and the target node have probabilistic distributions with a mean and variance, which are dominant parameters that have effects on the position error. A comparison study was conducted to determine the effects of the parameters of the target position in both a simulation and an experiment, showing rates of approximately 4% ~ 10%. These findings indicate that our simulation can work properly with the proposed method after assuming that the distance error is a Gaussian model.