Generalization and extension of the Wallace theorem (Q2880790)

The Wallace-Simson theorem states that the orthogonal projections \(A_1\), \(B_1\), \(C_1\) of any point \(P\) in the plane of a triangle \(ABC\) onto the sidelines \(BC\), \(CA\), \(AB\) are collinear if and only if \(P\) lies on the circumcircle of \(ABC\). Generalizations pertaining to other collinearities are obtained by \textit{O. Giering} in [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 207, 199--211 (1998; Zbl 1039.51008)]. These include collinearities of the points \(A_2\), \(B_2\), \(C_2\) and of the points \(A_3\), \(B_3\), \(C_3\), where \(A_2\), \(A_3\) are the points where the line \(AA_1\) meets \(CA\), \(AB\), respectively, and so on. Other types of generalizations are made by \textit{M. de Guzmán} in [Am. Math. Mon. 106, No. 6, 574--580 (1999; Zbl 1004.51021)], where the requirement that \(A_1\), \(A_2\), \(A_3\) be collinear is replaced by the requirement that the triangle \(A_1A_2A_3\) has a given area, and where the assumption that \(AA_1\), \(BB_1\), \(CC_1\) are perpendicular to the sidelines is replaced by the assumption that \(AA_1\), \(BB_1\), \(CC_1\) make an arbitrary angle \(\theta\) with these sidelines.NEWLINENEWLINEThe author of the paper under review pursues yet other types of generalizations in which \(AA_1\), \(BB_1\), \(CC_1\) make different angles with the sidelines of \(ABC\), and obtains several new results.