Triangles inscribed into triangles (Q1025799)

This article is concerned with area inequalities involving the following configuration \(\mathcal K\): On each side of a given triangle \(\Delta\) choose a point. Join these points with each other by straight line segments to form a triangle inscribed in the original one, thus producing a partition of \(\Delta\) into four triangles \(\Delta _0, \Delta _1, \Delta _2, \Delta _3 \) of respective areas \(F_0 , F_1, F_2, F_3,\) where \(\Delta _0\) denotes the central triangle inscribed in \(\Delta\). Then it is known that \[ F_0 \geq \min (F_1, F_2, F_3)\tag{1} \] and a similar inequality holds when \(F_i\)\,s are replaced by \(U_i\)\,s, representing the respective perimeters of triangles \(\Delta_i.\) These inequalities have been traditionally attributed to H. Debrunner, P. Erdős and E. Trost. In order to generalize inequality \((1)\), \textit{W. Janous} [Elem. Math. 61, No.~1, 32--35 (2006; Zbl 1135.51017)] asked for the best inequality of the type \[ F_0 \geq M_p (F_1, F_2, F_3),\tag{2} \] where \(M_p(x_1, x_2,\dots, x_n) = [(x_1^p + x_2^p + \cdots + x_n^p)/n]^{1/p}, \; \) \(- \infty \leq p \leq \infty \) is the \(p\)th-power mean of \(n\) positive real numbers \(x_1, x_2,\dots, x_n.\) In particular, when \(p = - \infty , 0, \infty\), after passing to a limit, \(M_p\) gives respectively the minimum, the geometric mean, and the maximum of \(x_i.\) Janous proved inequality \((2)\) when \(p = -1\) (the case of the harmonic mean) and conjectured that \(p = 1 - \log_23 \approx - 0.585\) is the best possible value since he found counterexamples to \((2)\) when \(p > 1 - \log_23.\) The purpose of the article under review is to prove inequality \((2)\) for \(p = 1 - \log_23 ,\) thus verifying the conjecture of Janous. In the proof the following result of J. F. Rigby is made use of: Four positive numbers \(F_i \geq 0\) can be realized as the areas of partitioning triangles \(\Delta_i\) in a configuration \(\mathcal K\) described above if and only if \(F_0^3 + (F_1 + F_2 + F_3) F_0^2 - 4 F_1 F_2 F_3 \geq 0\) [\textit{J. F. Rigby}, Math. Mag. 45, 113--116 (1972; Zbl 0237.50006)].