A theorem of Van Aubel, which first appeared in concerns the figure obtained by erecting squares on the sides of a quadrilateral, and considering various line segments such as the joins of the centres of opposite squares. In parts of the theorem are extended to the case of (i) four similar rhombi, and (ii) four similar rectangles erected on the sides of a quadrilateral. Further, these two results are described as in some sense ‘dual’ to each other. Here we show that they are in fact both special cases of a further extension of the original theorem (see Proposition 6, and the remarks that follow it), and we also find several extra squares in the original Van Aubel figure (Proposition 11).

Suppose the unit circle is divided into n > 1 equal arcs and points are placed at the ends of the arcs. Choose one of these points to be the ‘base point’ and draw chords connecting each of the other points to the base point. (See Figure 1 for the case n = 8.) Then the product of the lengths of these chords equals n (see Sichardt). In a related article, Eisemann considered the product of the lengths of the (nondegenerate) perpendiculars drawn from the centre of the unit circle to these chords.

In common with, I suspect, many people the author does not have access to the NAG library and so, when I was asked recently to calculate the value of the integral

correct to 10 decimal places my first reaction was to try several different calculators as well as several mathematical software packages. On doing so it was disappointing to find they either gave widely differing values such as 7.9065200767, 4.1317217452 or 0.9174196842 or an error message indicating that the method had not converged.

Problem 87.I in a recent Gazette concerns cyclic quadrilaterals.

We shall investigate generalisations of the problem by considering cyclic polygons and also points on a sphere, by distinguishing between sides and diagonals, and by looking at minima as well as maxima. The term ‘edge’ in the title is intended to encompass both sides and diagonals. It is interesting to see the wide variety of methods needed to solve the generalised problems.

Although a number of earlier researchers had used the geometric mean as a convenient statistic to summarise observational data, Gallon is usually credited with being the first to consider its sampling distribution. At Gallon’s request, in 1879 McAlister undertook a pioneering mathematical study, which eventually led to the modern large-sample theory. However, some sixty years elapsed before much attention was paid to small samples from particular parent distributions. Since about 1960, new techniques have made it possible to derive exact sampling distributions for a much wider class of parent distributions. Some work has been done on producing approximate general relationships between the moments of the parent distribution and those of the sample geometric mean but they are of very limited value for small samples and even now it is difficult to find any general description of how the distribution from which a sample is drawn will affect the distribution of its geometric mean and how this will vary with sample size.

You never know from where your next mathematical problem will come. I was surprised to find it, literally, in my own back yard. Recently, I was trying to determine the optimal way to water the lawn. I say ‘optimal’ because there are a couple of constraints that I try to minimise. First, I try to minimise the area of the wasted water. Water is wasted if it lands on the concrete that borders the lawn or if it lands on a region of the grass that has already been watered. Second, I try to minimise the number of times that I have to move the sprinkler. More often than not, I forget to move it anyway. So, it might as well be as a minimum.

This paper presents a number of theorems about hexagons whose three pairs of opposite sides are parallel. The first of these is a well-known result that the vertices of such a hexagon lie on a conic. Theorems 3 and 4 show how such conies are related to the Cevians of a triangle, and which Cevians lead to such conies being circles. When they are circles they are called Tucker circles. None of the results is at all obvious, yet it seems that some of the results presented here were known in the late nineteenth century or the early twentieth century. They seem to be of more than passing interest which should not get lost by the passage of time.