The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal, then the triangle is isosceles. In other words, if P is the incentre of triangle ABC, and if BP and CP meet the sides AC and AB at B′and C′, respectively, then

The human mind, by nature, is curious and enjoys dealing, both independently and competitively, with intellectual challenges. Throughout time, mathematical tasks, riddles, and puzzles have offered such challenges. In today's modern era, marketing companies and the media even offer prizes and rewards for the successful solution of puzzles, expanding the audience exposed to the various challenges. In addition, the styles, subjects and depths of the puzzles have become diversified. Experience gained in solving puzzles aids in developing reasoning abilities and deepening thought.

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

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

In an interesting Note [1] a method for constructing the directrices of a conic section using simple tools, viz. a pencil, a pair of compasses and a straight edge, was presented and it was wondered if the method might have been known before. From earlier days at school this author was aware that for a specific conic section (parabola, ellipse or hyperbola) displayed without reference on a piece of paper it was possible to determine not just the directrices but also the focal points and the usual axis set, again using just the simple tools. Others might also be aware of these facts, but if they are not so well known it might be of interest to present an argument in recipe format that will enable readers to appreciate a route to fathom the business, although there might well be other paths leading also to the same end-result, depending on the starting point.

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.

The purpose of this paper is to collect for the first time area formulas for all convex quadrilaterals that can be expressed in terms of their four sides alone. There are presently 23 such classes that we know of. Many mathematics teachers would probably find it challenging just to name that many different classes of quadrilaterals, and even more difficult to tell how to calculate their different areas. Several of these classes of quadrilaterals are included neither in MathWorld, Wikipedia nor in any other encyclopedia that we know of. The formulas can be sorted into four categories as either well known, a special case of another formula, less known, or previously unknown (at least to us). The last category only includes two formulas and we will prove these.

A Proof Without Words (PWW) contains data, the claim that is to be proved, and one or more diagrams, sometimes without anything else and in other cases with a few mathematical expressions, without any verbal justifications [1]. It is assumed that students and researchers who possess the related appropriate mathematical knowledge will view the drawings and the expressions, will be able to follow and justify each step in the proof and develop their own visual proof abilities. PWW is very much like a cartoon which contains a drawing with sometimes only a few words or sometimes no words at all and the observers are expected to understand the context or the projected message.

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.

A surveyor often encounters problems like Problem 1:

Determine the length of side AC of the right triangle ABC with the right angle at B. A lake intersecting the sides AC and CE prevents the direct measurement of their sides.

There is a long tradition of using geometry to solve problems from mechanics. Unfortunately this tradition is not practised much in schools and university any more.

With this exposition we would like to demonstrate how elementary properties of ellipses can be used to solve a problem related to the conical pendulum. The problem of the conical pendulum is to consider a mass attached to one end of a light inextensible string of length with the other end attached at the top of a vertical rod. The mass is moving about the rod in uniform circular motion in a horizontal plane. Given the angular velocity of the mass, the question is to determine the angle the string makes with the rod.

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.

Lack of information is generally considered a hindrance to inquiry. Surprisingly, a simple mathematical argument, relying on the Principle of Indifference, shows there are situations where the opposite holds. Even more surprisingly, this indifference allows one to guess, with a success rate greater than 50%, the outcome of a coin toss or any other experiment having two equiprobable outcomes. The scheme is based on work by American statistician David Blackwell (1919–2010) and a principle of mathematical probability attributed to Swiss mathematician Jakob Bernoulli (1655–1705).

In the present paper, we show that the point of intersection of the bimedians of a cyclic quadrilateral belongs to the nine-point circle (Euler’s circle) of the triangle with one vertex at the point of intersection of the quadrilateral’s diagonals and the other vertices at the points of intersection of the extensions of its pairs of opposite sides.

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).

This paper presents some new theorems about the Pascal points of a quadrilateral. We shall begin by explaining what these are.

Let ABCD be a convex quadrilateral, with AC and BD intersecting at E and DA and CB intersecting at F. Let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. By using Pascal’s theorem for the crossed hexagons EKNFML and EKMFNL and which are circumscribed by ω, the following results can be proved [1]:–

1. (a) NK, ML and AB are concurrent (at a point P internal to AB)

2. (b) NL, KM and CD are concurrent (at a point Q internal to CD)

A number of methods are available for the solution of cubic equations. Amongst these, and possibly less well known than those based on algebraic or trigonometrical procedures, is an elegant graphical solution devised by Lalanne [1]. His approach and the resulting nomogram are appraised in this paper.

The general cubic equation in the variable X

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]}}} $.