On the number of circles determined by \(n\) points in the Euclidean plane (Q1320453)

Let \(P=\{p_ 1,p_ 2, \dots, p_ n\}\) be a set of \(n\) points in the Euclidean plane and \(S\) be the set of connecting lines, which join these points, and \(C\) be a set of circles determined by points of \(P\). The authors formulate their results in the following two theorems. Theorem 1. Let \(P\) be a set of \(n \geq 4\) points in the Euclidean plane, not all on a circle or a straight line. Let \(p_ j\) be an arbitrary point of a set \(P\). Then \(P\) determines at least \(15(n-1)/133\) circles containing exactly three points of \(P\), one of which is \(p_ j\). Theorem 2. Let \(P\) satisfy the hypotheses of Theorem 1. Then \(k_ 3 \geq 5n(n-1)/133\), where \(k_ 3\) is the cardinality of the set of circles \(c \in C\) with \(| c \cap P |=3\).