A generalization of the Conway circle (Q2871655)

Associated to every triangle \(ABC\) is a hexagon \(\mathbb{H}=A_cC_aB_aA_bC_bB_c\) obtained by defining \(A_c\) to be the point on the extension of \(BC\) for which \(CA_c=BA\), and defininig the other points by permuting. Letting \(\mathcal{I}\) be the incenter of \(ABC\), the point \(A_c\) can also be defined by letting \(A^c\) be the point where the line \(B\mathcal{I}\) meets the line drawn from \(A\) parallel to \(BC\) and then defining \(A_c\) to be the point for which \(A^c AC A_c\) is a parallelogram. In view of this latter definition, it is natural to denote the hexahedron \(\mathbb{H}\) by \(\mathbb{H}_{\mathcal{I}}\) and to define \(\mathbb{H}_P\) for every point \(P\) inside \(ABC\) similarly. NEWLINENEWLINENEWLINEIt is known that the vertices of \(\mathbb{H}_{\mathcal{I}}\) lie on a circle, known as the Conway circle. The paper under review proves that for every point \(P\) inside \(ABC\), the vertices of \(\mathbb{H}_P\) lie on a conic, and it explores properties of the configuration of \(\mathbb{H}_P\). Among other things, it is proved that the triangle formed by the lines \(B_aC_a\), \(C_bA_b\), \(A_cB_c\) and the triangle formed by the lines \(B_cC_b\), \(C_aA_c\), \(A_bB_a\) are perspective with \(ABC\). Denoting the point of perspectivity by \(P^*\), it is observed that \(\mathcal{I}^*\) is the Gergonne point, leaving other properties of the map \(P \mapsto P^*\) for further investigation.