Kites possess complex flying characteristics that must be understood to ensure efficient operation in airborne wind energy applications. The flight dynamics of a six-degrees-of-freedom three-line delta kite with full tether dynamics through a spring-mass model is investigated. Results reveal nonlinear behaviours that agree with prior analyses on simpler models of single-tether systems with constrained dynamics. These include the co-existence of multiple steady-state solutions at low airspeeds and a pendulum-mode instability that puts the kite in an ‘automatic' figure-of-eight trajectory without any active control input. In the three-line system studied, this figure-of-eight trajectory is identified to be a stable limit cycle, which is influenced by the lateral distance between the left and right tether anchor points on the ground station. Increasing this distance stabilises the pendulum mode, which destroys the limit cycle and puts the kite in a stable trim condition. However, doing so results in excessive lateral stability, thereby reducing the manoeuvrability of the kite. This finding highlights the impact of ground station design on kite flight dynamics and underlines a trade-off between stability and manoeuvrability.