A generalized Seebach's theorem (Q464806)

In his paper in [Elem. Math. 42, No. 5, 132--139 (1987; Zbl 0702.51002)], \textit{K. Seebach} gave a very complicated proof of the fact that if two triangles \(ABC\) and \(A'B'C'\) are given, then there exists inside \(ABC\) a unique point \(P\) whose cevian triangle \(A_PB_PC_P\) is directly similar to \(A'B'C'\). The obscurity of this result is evidenced by the appearance of [Problem 10358*, Am. Math. Mon. 101, 76 (1994)] by \textit{J. Huanxin} asking for a proof in the special case when \(A'B'C'\) is equilateral, and much later of a solution, again very complicated, in [Solution to Problem 10358*, Am. Math. Mon. 104, 567--570 (1997)] by the single solver \textit{D. Goering}. Huanxin's problem and Goering's solution are quoted on page 267 of \textit{C. Kimberling}'s encyclopedic work [Congr. Numerantium 129, 1--285 (1998; Zbl 0912.51009)], where \(P\), in this special case, is denoted by \(X_{370}\) and is referred to as the equilateral cevian triangle point of \(ABC\). Several years later, this reviewer gave a much simpler and almost computation-free proof of Seebach's theorem in [\textit{M. Hajja}, Am. Math. Mon. 113, No. 5, 443--447 (2006; Zbl 1165.51007)]. In all of the aforementioned references, the point \(P\) was confined to lie \textit{inside} \(ABC\). The author of the paper under review generalizes Seebach's theorem to include points that lie anywhere in the plane of \(ABC\). He shows that there exist one to three points whose cevian triangle is positively oriented and directly similar to \(A'B'C'\), and he shows that the trilinear coordinates of such points can be expressed in terms of the zeros of a certain cubic. He proves a similar result for negatively oriented triangles. Then he studies in detail the situation when \(A'B'C'\) is \(ABC\) itself, and he also exhibits examples of triangles \(ABC\) for which the center \(X_{370}\) is not Euclidean constructible.