Geometric circle fitting is a fundamental task in scientific and engineering research. In this contribution, a parameter-free approach for weighted geometric circle fitting is proposed based on the iteratively linearized model with condition equations. Furthermore, the corresponding bias analysis is investigated by the classical theory of nonlinear least-squares. The simulated results show that: 1) Our new approach obtains the optimal solution in various weighted cases and its convergence behavior is more stable than the classical parametric method; 2) The biases caused by nonlinearity can be effectively removed in different situations. Finally, the measured coordinates of an archaeological site in Britain, the Brogar Ring, are adjusted and the results are analyzed.