Antirhombi (Q2878865)

Let \(Q=ABCD\) be a plane quadrilateral, and let the lengths of sides \(AB\), \(BC\), \(CD\), and \(DA\) be denoted by \(a\), \(b\), \(c\), and \(d\), respectively. A theorem of Pithot states that \(Q\) admits an incircle (i.e., a circle that touches the sides of \(Q\) internally) if and only if \(a+c=b+d\). When this happens, \(Q\) is said to be circumscriptible (or circumscribed), and the center of the incircle is called the incenter, and is denoted by \(\mathcal{I}\). The lengths of the tangents from \(A\), \(B\), \(C\), and \(D\) to the incircle are denoted by \(x\), \(y\), \(z\), and \(w\), respectively. The centroid of \(Q\) is defined to be the center of mass of four equal masses placed at the vertices, and is denoted by \(\mathcal{G}_0\).NEWLINENEWLINE The paper under review starts with a classification of all quadrilaterals in which \(\mathcal{I}\) and \(\mathcal{G}_0\) coincide. This clasification is taken from \textit{D. Grinberg} in [``Circumscribed quadrilaterals revisited'', \url{http://www.cip.ifi.lmu.de/~grinberg/CircumRev.pdf}, \url{http://web.mit.edu/~darij/www/geometry2.html}], and it states that circumscriptible quadrilaterals \(Q\) for which \(\mathcal{I} = \mathcal{G}_0\) are precisely the (i) rhombi, (ii) isosceles trapezoids (necessarily with \(a+c=b+d\)), and (iii) quadrilaterals with no parallel sides and with \(x+z=y+w\). After giving another proof, the author devotes the rest of the paper to exploring other interesting properties of quadrilaterals of the type described in (iii) above, and which he calls, following Baloglou, antirhombi.NEWLINENEWLINEThe author of the paper under review is apparently unaware of the paper by \textit{A. Al-Sharif} et al. [Result. Math. 55, No. 3--4, 231--247 (2009; Zbl 1191.51004)], where the authors established (in Theorem 5.6) the characterization given above, as well as similar characterizations of quadrilaterals in which other pairs of centers coincide. These include the Fermat-Torricelli point \(\mathcal{F}\), the center of mass \(\mathcal{G}_1\) of rods of the same uniform density placed on the sides of \(Q\), and the center of mass \(\mathcal{G}_2\) of a lamina of uniform density placed over \(Q\).NEWLINENEWLINEIt is worth mentioning that the degree of regularity implied by the coincidence of two or more centers of a geometric figure has recently attracted attention. For the coincidence of centers for triangles, see the papers by \textit{M. Hajja} in [Int. J. Math. Educ. Sci. Technol. 32, No. 1, 21--36 (2001; Zbl 1011.51006)] and by \textit{S. Abu-Saymeh} and \textit{M. Hajja} in [Forum Geom. 7, 137--146 (2007; Zbl 1162.51311)]. For tetrahedra and higher dimensional simplices, see the papers by \textit{A. L. Edmonds} et al. in [Beitr. Algebra Geom. 46, No. 2, 491--512 (2005; Zbl 1093.51014)] and in [Result. Math. 47, No. 3--4, 266--295 (2005; Zbl 1084.51008)], and the paper by \textit{M. Hajja} in [Result. Math. 49, No. 3--4, 237--263 (2006; Zbl 1110.52014)].