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Given a set ofndemand points with weightWi,i= 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An Ω(nlogn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.