Two new proofs of Goormaghtigh's theorem (Q2848761)

A theorem by Réne Goormaghtigh published in 1941 says: Let \(\Delta=ABC\) be a triangle of the (real) Euclidean plane with the circumcenter \(O\) and assume that \(\Delta\) is not isoceles rectangular. If \(A'\) is a point on \(OA\), \(B'\) one on \(OB\), and \(C'\) one on \(OC\) such that NEWLINE\[NEWLINE \overline{OA'}/\overline{OA}=\overline{OB'}/\overline{OB}=\overline{OC'}/\overline{OC}=k NEWLINE\]NEWLINE is a positive real and let \(a_p\), \(b_p\), and \(c_p\) be the perpendiculars to \(OA\) at \(A'\), to \(OB\) at \(B'\), and to \(OC\) at \(C'\), respectively, then the three points \(a_p\cap\,BC=:X\), \(b_p\cap\,CA=:Y\), \(c_p\cap\,AB=:Z\) are collinear. (Compare also \textit{K. L. Nguyen} [Forum Geom. 5, 17--20 (2005; Zbl 1078.51507)]).NEWLINENEWLINE The authors present two synthetic proofs of Goormaghtigh's theorem, one based on the Desargues theorem, the other on the converse of the Menelaus theorem. Moreover, the authors derive some consequences of Goormaghtigh's theorem; for instance: The circumcircles of the four triangles \(AYZ\), \(BZX\), \(CXY\) and \(\Delta\) pass through a common point. Finally, the authors examine the dependence of the considered configuration on the homothety coefficient \(k\).