An extremal problem for inscribed triangles (Q2461424)

\textit{H. Debrunner} [Aufgabe 260, Elem. Math. 11, 20 (1956)] first asked in print for the proof of the following problem, attributed to \textit{E. Trost} and \textit{P. Erdős}: ``If triangle \(XYZ\) is inscribed in triangle \(ABC\) -- with \(X, Y, Z\) on sides \(BC, CA,\) and \(AB\) -- then the smallest area (perimeter) among the triangles \(BXZ, CXY, AYZ\) is less than or equal to the area (perimeter) of triangle \(XYZ\), with equality if and only if \(X, Y\), and \(Z\) are the midpoints of the sides \(BC, CA,\) and \(AB\).'' The first proof to appear, for the area case only, was that of \textit{A. Bager} [Elem. Math. 12, 43 (1957)], leaving the perimeter case open. It reappeared as proposed problem, in the area version, in \textit{J. Rainwater (pseudonym)} [Problem 4908, Am. Math. Mon. 67, 479 (1960)], with one solution, by \textit{P. H. Diananda} [Problem 4908, Am. Math. Mon. 68, 386--387 (1961)], providing a sharper inequality. \textit{E. Morgantini} [Rend. Sem. Mat. Univ. Padova 30, 245--247 (1960; Zbl 0099.15503)] provided another proof for the area case, whereas \textit{G. C. Citterio} [Period. Mat., IV. Ser. 40, 41--50 (1962; Zbl 0103.38002)] provided a proof of the same case using only affine notions. The perimeter case also appeared as a problem proposed by \textit{A. Oppenheim} [Nabla (Bull. Malayan Math. Soc.) 7, 165 (1960)], with a solution by \textit{L. A. G. Dresel} appearing in the same journal [8, 97 (1961)], where the theorem is extended to the radii of the inscribed circles of the four triangles involved. It reappears as [Problem 4964, Am. Math. Mon. 68, 384 (1961)], for the perimeter case only, proposed by \textit{E. Trost} and \textit{A. Bager}, with solution by \textit{R. Breusch} in [Am. Math. Mon. 69, 672--674 (1962)]. Another proof for the perimeter case was given by \textit{H. T. Croft} [Math. Gaz. 49, 45--49 (1965; Zbl 0196.52802)]. \textit{E. Szekeres} [Elem. Math. 22, 17--18 (1967; Zbl 0149.17801)] provided very simple proofs for both the area and the perimeter forms of the problem, the area one being a purely affine proof. Diananda's inequality was sharpened by \textit{J. F. Rigby} [Math. Mag. 45, 113--116 (1972; Zbl 0237.50006)], and a like-minded one was provided by \textit{O. Bottema} [Math. Mag. 47, 34--36 (1974; Zbl 0286.50005)]. In its area form, the problem was generalized to triangles inscribed in convex Jordan curves by \textit{S. L. Chow} [Generating and drawing area-proportional Euler and Venn diagrams. Ph.D. Thesis, Department of Computer Science, University of Victoria, Victoria, BC (2007), Lemma 3.4.1 in Appendix A3.4]. In the paper under review, the author attempts to find conditions a function \(\mu\) of the sides of a triangle, symmetric in its three arguments \(a, b, c\), has to satisfy, to allow for the same conclusion as in the Trost-Erdős cases (in which \(\mu_1(a,b,c)\) equals the area of \(\triangle ABC\), and \(\mu_2(a,b,c)=a+b+c\)), if we replace the words area or perimeter with \(\mu\), to hold. He proves a theorem valid for all monotone \(\mu\), in the sense that \(\mu(\Delta_1)<\mu(\Delta_2)\) whenever triangle \(\Delta_1\) is included in triangle \(\Delta_2\), and uses it to provide somewhat uniform proofs for the Trost-Erdős cases and for the Dresel case (in which \(\mu(a,b,c)\) is the radius of the circle inscribed in \(\triangle ABC\)). He shows that the monotonicity condition is necessary for \(\mu\), as the theorem no longer remains true if \(\mu(a,b,c)\) is the radius of the circumscribed circle of \(\triangle ABC\), and offers a conjecture for the necessary and sufficient condition \(\mu\) needs to satisfy to achieve the Trost-Erdős conclusion. The author was under the impression that, at least the perimeter case of the problem, was open, being stated as a conjecture in \textit{N. D. Kazarinoff} [Analytic inequalities. New York: Holt, Rinehart and Winston (1961; Zbl 0097.03801)], republished [Analytic inequalities. Unabridged slightly revised republication of the 1961 original. Mineola, NY: Dover Publications (2003; Zbl 1085.26005)].