This paper presents a novel Lagrangian algorithm, the Lagrangian Eddy Boundary Delineation Algorithm (LEBDA), designed for determining the horizontal boundaries of mesoscale quasi-stationary eddies. In contrast to traditional Eulerian methods employed for similar tasks, LEBDA minimizes abrupt boundary changes, a critical advantage particularly relevant for the analysis of long-lived structures. A key feature of the algorithm is the use of passive tracer trajectories to construct R-contours, enabling accurate identification of both the eddy core and its periphery, while effectively filtering out transient perturbations. A comparison with the Automated Mesoscale Eddy Detection Algorithm (AMEDA), using the Lofoten Vortex as a case study, demonstrated that LEBDA provides a more physically consistent and temporally smooth delineation of the eddy boundary. Unlike AMEDA, which tends to overestimate eddy sizes (by up to 1.5 times), LEBDA exhibits a gradual and realistic evolution of boundaries. Analysis of the Lofoten Vortex parameters using LEBDA revealed a relationship between lobe formation, perimeter variations, interaction with the surrounding flow, and the influence of convection on the eddy's shape and stability.