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In this paper, we propose a novel remote user authentication scheme based on the geometric property of the Euclidean plane for wireless networks. We combine a user's biometrics information with a password to enhance the level of the security. We demonstrate that the proposed scheme satisfies the following security properties: anonymity, resistance to forgery attacks and replay attacks, fast error detection, resistance to off-line password guessing attacks, resistance to server overloading attacks, and users can choose and change their passwords easily.