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Fitting noisy measurements to a circle is a classic statistical estimation problem. In this paper, we make two contributions to the study of this problem. First, we propose a novel formulation of the maximum likelihood (ML) estimator for identifying the center and radius of the circle from noisy measurements. This new estimator uses the unknown true values of the measurement points as the nuisance parameter to obtain an exact ML formulation. We then examine the Karush-Kuhn-Tucker (KKT) conditions for the optimum solution to the ML estimator. We show analytically that this new estimator is in fact equivalent to the well-known least squares (LS) form of the circle fitting problem. Second, from the insights gained in deriving the optimum solution, a computationally simple circle fitting algorithm based on greedy search is proposed. Performance results are given to illustrate the performance of the proposed algorithm.