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Quadratic Bezier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic Bezier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing G1 quadratic Bezier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the G1 quadratic Bezier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs.