Crossed ladders and power means ∗

Given to positive numbers a and b, we show how we easily can construct the power mean Pk(a, b) of order k for the cases k = −2, −1, −1/2, 0, 1/2, 1 and 2. This is done by observing that the power means correspond to certain distances in the crossed ladders problem. The so-called crossed ladders problem, of unknown origin, has been discussed in the literature at least since 1895 (see [3, p. 62-64]). Consider two ladders leaning against two vertical walls at heights a and b above the floor and with distance d between the walls (see Figure 1). These ladders cross each other at a point with distance c above the floor. The problem is devoted to the determination of various types of relations between the above quantities. The beauty of the problem is that at first it looks very simple, but one soon realizes that many aspects of it may be rather difficult, as seen from the discussions in e.g. [1], [5], [6] and [7]. In this paper, however, we will focus on some simple aspects of the crossed ladders problem which, as far as we know, have not been offered much attention in the literature, namely its relation to the basic power means. Power means have fascinated mathematicians since antiquity. For two positive numbers a and b the power mean Pk (with equal weights) of order k is defined by Pk = ⎨⎩ 3 a+b 2 ́ 1 k , if k 6= 0, √ ab, if k = 0. The most common power means are the arithmetic mean A = P1, the geometric mean G = P0 and the harmonic mean H = P−1, given by the formulae A = a+ b 2 , G = √ ab, H = 2ab a+ b , respectively. By using the inequality a + b ≥ 2ab, which follows by the fact that (a− b) ≥ 0, it is easily seen that H ≤ G ≤ A. More generally, it follows ∗To appear in Elem. Math. in 2008