Andrew Palfreyman’s article [1] reminds us of the result (1)

$${\rm{ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}\,2{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ 3 = }}\,\pi {\rm{, }}$$

A convex quadrilateral ABCD is called circumscriptible or tangential if it admits an incircle, i.e. a circle that touches all of its sides. A typical circumscriptible quadrilateral is depicted in Figure 1, where the incircle of ABCD touches the sides at, Aʹ, Bʹ, Cʹ and Dʹ. Notice that labellings such as AAʹ = ADʹ = a are justified by the fact that two tangents from a point to a circle have equal lengths (a, b, c and d in Figure 1 are called tangent lengths). This simple fact also implies that if x, y, and z are the angles shown in the figure, then x = y. In fact, if AD and BC are parallel, then x = y = 90°. Otherwise, the extensions of AD and BC would meet, say at Q, with QDʹ. Hence x = y. Thus x = y in all cases, and sin x = sin y = sin z. We shall use this observation freely. Also we shall denote the vertices and vertex angles of a polygon by the same letters, but after making sure that no confusion may arise.

Recently, Gerhard J. Woeginger [1] gave a survey on the interesting history of results on equiangular n-vertex polygons with edge lengths in arithmetic progression. Such a polygon exists if, and only if, n has at least two distinct prime factors.

Three familiar properties of a parabola are that it is the locus of points that are equidistant from the focus and the directrix, that it can be created by an intersection of a plane and a cone, and that incoming rays parallel to the axis are reflected to a single point. The first two are often used as definitions, and the third may be used as an alternative definition or characterisation.

The Chinese Ring puzzle consists of a long loop with a handle on one end and nine rings entwined on the loop (see Figure 1).

What are the units, irreducibles, and primes of the ring ${\mathbb{Z}}\sqrt n $, the set of all numbers $a + b\sqrt n $ where a and b are integers and n is a fixed positive square-free integer? In the ring ${\mathbb{Z}}$, primes and irreducibles are synonymous and its units are ±1. ${\mathbb{Z}}\sqrt n $ is wilder, and our modest goal here is to catalogue all such numbers for ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $, where a and b range from 0 to 10; the result appears in Figure 1. Here are a few teasers that may induce a reader to read on: $3 + \sqrt {10} $ is a unit; 2, 3, 5, and 7 are irreducibles, but not 31; and 7 is the least positive integer that is prime in both ${\mathbb{Z}}$ and ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $.

Consider the problem of proving that, for any positive numbers x, y and z,

$${\rm{9 (}}{{\rm{x}}^{\rm{3}}}{\rm{ + }}{{\rm{y}}^{\rm{3}}}{\rm{ + }}{{\rm{z}}^{\rm{3}}}{\rm{) }} \ge {\rm{ (x + y + z}}{{\rm{)}}^{\rm{3}}}{\rm{ }}{\rm{.}}$$

This is an example of a type of inequality that frequently occurs in Olympiad-style problems, [1]. These problems may involve symmetric functions of more or fewer variables than the three used here. However, three variables are commonly used and appear to give appropriately difficult problems without making excessive computational demands.

Which is your favourite formula in triangle geometry? Mine is definitely the formula for the area of triangle OIH . It is well known that the perpendicular bisectors to the sides of any triangle are concurrent at a point O (centre of the circumcircle), that the angle bisectors to the vertex angles are concurrent at a point I (centre of the incircle), and that the altitudes are concurrent at a point H. If the triangle is not isosceles, then these three points are all different and uniquely determine a new triangle OIH (see Figure 1), whose area can be expressed in terms of the sides a, b, c of the original triangle. I derived such a formula 20 years ago, and later found out that it had been studied a century earlier.

When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [1, p. 171f] (see also [2, p. 59], [3, p. 222f] and [4, p. 42ff]).