Variance on topics of plane geometry (Q2856605)

The book under review is actually a collection of 21 separate short articles in plane Euclidean geometry. Nineteen of these articles are written by the two authors of the book, and the remaining two jointly with a third author, \textit{C.~Coandă}. Among other things, the papers deal with quasi-isogonal triangles, nedians (or generalized medians), Carnot circles, polar of a point, anti-bisectors, anti-heights, formulas for distances between remarkable points of a triangle, Euler's line, applications of Desargues' theorem, mixt-linear circles, and orthohomological, orthological, bi-orthological, and tri-homological triangles.NEWLINENEWLINENEWLINENEWLINEThe papers in the book could have been submitted to journals devoted, partly or fully, to plane geometry, such as \textit{Forum Geom.}, \textit{J. Geom. Graphics}, and \textit{Internat. J. Geom.} Referees' feedback would then have resulted in versions that are more polished language-wise and also content-wise, as illustrated by the examples below. Writing things with this great haste can only make great waste, and reduce readership.NEWLINENEWLINENEWLINENEWLINEFor example, Article \# 1 proves the existence of triangles that the authors call \textit{quasi-isogonal}. These are triangles \(ABC\), having an obtuse angle at \(B\), in which the altitude \(AP\) and the median \(AM\) have the property that \(\angle CAM = \angle BAP\). A referee would have very likely remarked that this immediately follows from starting with any triangle \(APM\) having a right angle at \(P\) and letting two points \(B\) and \(C\) start at \(M\) and move away from each other on the line \(PM\) keeping the distances \(BM\) and \(MC\) equal. It would then follow from trivial continuity considerations that there is a unique location of \(B\) and \(C\) for which \(ABC\) is quasi-isogonal. A referee would have also suggested characterizing quasi-isogonal triangles in terms of their side-lengths, and exploring whether such characterizations have geometric interpretations.NEWLINENEWLINENEWLINENEWLINEArticle \#2 starts by calling a cevian of triangle a \textit{nedian} if it divides the respective side in the ratio \(1 : n-1\), and then studies triangles whose side lengths are the lengths of the three nedians and also the triangles that are enclosed by the three nedians. It then studies issues pertaining to similarity of certain triangles in the configuration, the Brocard points, and the so-called \textit{coefficient of deformation}. Here again, a referee would have noticed that these issues have already been encountered and studied in full detail, in the context of studying the convergence of certain iterations, in two papers by \textit{M. Hajja} in [Result. Math. 54, No. 3--4, 289--299 (2009; Zbl 1183.51006)] and [J. Geom. 96, No. 1--2, 71--79 (2010; Zbl 1204.51020)]. There, a cevian that divides the respective side in the ratio \(s : 1-s\), \(s \in \mathbb R\), is called an \(s\)-median, the triangle whose side lengths are the lengths of the three \(s\)-medians the \(s\)-median triangle, and the triangle enclosed in the three \(s\)-medians the \(s\)-Routh triangle. The coefficient of deformation is essentially the norm of the shape function introduced in the aforementioned papers of Hajja.NEWLINENEWLINENEWLINENEWLINEOne expects that similar improvements could have been made on the other articles, had the authors chosen to be less hasty in sending their works to print.NEWLINENEWLINENEWLINENEWLINEMany of the comments about typos, typesetting, and language and style that were made by the reviewer of the authors' earlier book [The geometry of homological triangles. Columbus, OH: The Education Publisher (2012; Zbl 1298.51004)] apply to this book as well. Lists of references in the papers could certainly have been expanded.