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• 1 Bolyai Institute, University of Szeged & Department of Mathematics and Statistics, Auburn University 1 Aradi vértanúk tere, H-6720 Szeged, Hungary & 221 Parker Hall, Auburn, AL 36849, U.S.A 1 Aradi vértanúk tere, H-6720 Szeged, Hungary & 221 Parker Hall, Auburn, AL 36849, U.S.A
• 2 Department of Mathematics and Statistics, Auburn University & MTA Alfréd Rényi Institute 221 Parker Hall, Auburn, AL 36849, U.S.A. & , Reáltanoda u. 13--15, Budapest, Hungary 221 Parker Hall, Auburn, AL 36849, U.S.A. & , Reáltanoda u. 13--15, Budapest, Hungary
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## Summary

The central problem of this paper is the question of denseness of those planar point sets \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document}, not a subset of a line, which have the property that for every three noncollinear points in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document}, a specific triangle center (incenter (IC), circumcenter (CC), orthocenter (OC) resp.) is also in the set \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document}. The IC and CC versions were settled before. First we generalize and solve the CC problem in higher dimensions. Then we solve the OC problem in the plane essentially proving that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document} is either a dense point set of the plane or it is a subset of a rectangular hyperbola. In the latter case it is either a dense subset or it is a special discrete subset of a rectangular hyperbola.

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