The \(f\)-belos (Q2871644)

The Archimedean arbelos consists of three semicircles having collinear diameters \(AP\), \(PB\), \(AB\) and drawn on the same side of \(AB\). Replacing these semicircles by parabolic segments having \(AP\), \(PB\), \(AB\) as latus rectums, as done by \textit{J. Sondow} in [Am. Math. Mon. 120, No. 10, 929--935 (2013; Zbl 1291.51018)], one obtains what we now call the ``parbelos''.NEWLINENEWLINEThe paper under review introduces yet another variant of the arbelos configuration that encompasses both the arbelos of Archimedes and the parbelos of Sondow. Starting with a fairly smooth positive function \(f : [0,1] \to \mathbb R\), with \(f(0)=f(1)=0\), and an arbitrary point \(P=(p,0)\) with \(0 < p < 1\), the author considers the configuration consisting of the graph of \(f\) and two more graphs on \([0,p]\) and \([p,1]\) that are similar to that of \(f\), i.e., the graphs of the functions \(g : [0,p] \to \mathbb R\) and \(h: [p,1] \to \mathbb R\) defined by NEWLINE\[NEWLINEg(x) = p~ f\left( \frac{x}{p} \right),~~ h(x) = (1-p)~ f\left( \frac{x-p}{1-p} \right).NEWLINE\]NEWLINE It is easy to see that when \(f(x) = \sqrt{x-x^2}\), one obtains the original Archimedean arbelos, and when \(f(x) = x - x^2\), one obtains Sondow's parbelos. The author then studies the configuration and obtains several interesting properties.NEWLINENEWLINEAssuming that the tangent lines to the graph of \(y=f(x)\) at \(0\) and \(1\) are not parallel, meeting say at \(U\), and letting the tangent lines to the graphs of \(y=g(x)\) and \(y=h(x)\) at \(P\) meet the sides \(UA\) and \(UB\) at \(E\) and \(F\), we easily see that \(UEPF\) is a parallelogram. In view of this, one may reverse the process and start with a triangle \(UAB\), a point \(P\) on \(AB\), and two line segment \(PE\) and \(PF\) that are parallel to the sides \(UA\) and \(UB\) and meeting them at \(E\) and \(F\). Then one can draw a graph that is tangent to triangle \(UAB\) at \(A\) and \(B\) and two similar graphs on \(AP\) and \(BP\). If these graphs are chosen to be parabolas, then one obtains a generalized parbelos configuration that would coincide with the one studied by Sondow when \(UA=UB\) and \(\angle AUB = 90^{\circ}\). This answers the question raised at the end of the review of Sondow's afore-mentioned paper. It is true that given any triangle \(UAB\), there exists a unique parabola that is tangent to the sides \(UA\), \(UB\) at \(A\), \(B\). The interested reader can find an explicit construction in Construction 49 of \textit{T. H. Bagles}'s [Constructive geometry of plane curves. London: \(8^{\circ}\) (1886; JFM 18.0651.04)]