In a paper by A. Bialostocki and D. Bialostocki [Forum Geom. 11, 9–12 (2011; Zbl 1213.51012)], the authors let a point \(P\) move on the internal bisector \(L\) of \(\angle A\) in triangle \(ABC\) and they proved that the ratio \(PB/PC\) attains its maximum and minimum at the incenter and the \(A\)-excenter, respectively. Later, a considerably shorter proof of this result was given by this reviewer M. Hajja in [Math. Gaz. 96, Note 96.44, 315–317 (July 2012)]. The authors of the paper under review prove a more general result: they start with any line \(L\) and any two points \(B\) and \(C\), and they locate the points \(P\) on \(L\) at which the ratio \(PB/PC\) attains its maximum and minimum. The proof is short and uses calculus. Apparently, they are neither aware of the aforementioned paper of M. Hajja, nor aware of the recent short and calculus-free note of G. Nicollier in which the same results are proved. Nicollier’s note is expected to appear in [Math. Gaz. (March 2016)].
It is natural to wonder whether the result in this paper is indeed stronger than the earlier results. Thus given a line \(L\) and two points \(B\) and \(C\), one asks whether there exists a point \(A\) on \(L\) such that \(L\) is the internal bisector of \(\angle A\) of triangle \(ABC\). It turns out that, except in few extreme cases, \(L\) is indeed the internal (respectively, external) bisector of \(\angle A\) of some triangle \(ABC\) if \(L\) crosses the line segment \(BC\) internally (respectively, externally). This should not reduce the value of this paper in any way.