The following theorem was conjectured by Erdős and Purdy: Let P be a set of \(n>4\) points in general position in the plane. Suppose that R is a set of points disjoint from P such that every line determined by P passes through a point in R. Then \(|R| \ge n\). In this paper we give a very elegant and elementary proof of this, being a very good candidate for the “book proof” of this conjecture.
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We thank Gábor Tardos for a clever remark that helped to shorten the presentation of the proof.
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Dedicated to the memory of Branko Grünbaum.
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R. Pinchasi: Supported by ISF Grant (Grant No. 409/16). A. Polyanskii: Supported in part by the Leading Scientific Schools of Russia through Grant No. NSh-6760.2018.1.
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Pinchasi, R., Polyanskii, A. A One-Page Solution of a Problem of Erdős and Purdy. Discrete Comput Geom 64, 382–385 (2020). https://doi.org/10.1007/s00454-019-00139-1