Several problems that were composed and/or solved by the tenth century Islamic mathematician Abu Sahl al-Kuhi reached us via the writings of Abd al-Jalil al-Sijzi, another tenth century Islamic mathematician who, according to [1], was presumably a student of al-Kuhi’s. Twelve of these (sets of) problems and theorems are discussed in [1] and are referred to as The Fragments. The theorem of al-Kuhi alluded to in the title is Fragment #9, which is presented below, together with Fragment #5 and their proofs, in Section 2. Section 3 is devoted to the generalisation referred to in the title. Section 4 describes a relation to angle trisection and Sections 5 and 7 a relation too a configuration of Serenus. Section 6 contains a speculation on what motivated Fragment #9.

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal then the corresponding sides have equal lengths. That is to say if P is the incentre of ΔABC and if BP and CP meet the sides AC and AB at B′ and C′, respectively, then

An elegant proof of this theorem appeared in [1] and is reproduced in [2].

This article, like our previous one [1], combines known and new characterisations of parallelograms. Both can be thought of as additions to Martin Josefsson’s series on ‘characterisations of’ and ‘properties of’ various types of quadrilaterals – a series that does not include parallelograms. Josefsson’s publications can be found listed in [2], [3] and [4]. For the importance of characterisations in geometry, see [5].

Several years ago, Professor Martin Josefsson told the first-named author, in a private communication, that his ‘characterisations of’ and ‘properties of’ series of papers that comprised tangential, extangential, bicentric, orthodiagonal, equidiagonal, and bisect-diagonal quadrilaterals, and rhombi and trapezoids, did not include parallelograms because the many characterisations of these figures are well known and are easily accessed via the net, and that nothing interesting can be added. In this Article, we present some characterisations of parallelograms that are very likely unknown and that will hopefully appeal to the readers of the Gazette. Other interesting characterisations that we have obtained and that could not find their place here are intended to form the material of another Article.

Figure 1 shows a triangle ABC with the midpoints A′, B′ and C′ of its sides. The line segments AA′, BB′ and CC′ are called the medians, and the point G of their intersection the centroid. The line segments AG, BG and CG will be called, for lack of a better name, the semi-medians. It is interesting that the medians of any triangle can serve as the side lengths of some triangle. This property of the medians is referred to as the median triangle theorem in [1, §473, page 282], and is discussed, together with generalisations to tetrahedra and higher dimensional simplices, in [2].

Following Euler, we denote the side lengths and angles of a triangle ABC by a, b, c, A, B, C in the standard order. Any line segment joining a vertex of ABC to any point on the opposite side line will be called a cevian, and a cevian AA′ of length t will be called long, strictly long, or balanced according as t ≥ a, t > a or t = a. If A′ lies strictly between B and C, AA′ is called an internal cevian. This convention regarding cevians is not universal, and it is, for example, in a heavy contrast with that in [1, p. 73].

Figure 1 below shows a triangle ABC and three cevians AX, BY and CZ with intersection points

This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.

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 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

This Article is inspired by a problem that appeared recently in the American Mathematical Monthly, namely 11972 in [1, p. 369]. The problem asks the readers to prove that if r is the inradius of a tetrahedron, and if r1, r2, r3, r4 are its exradii, then

By taking r = r1 = r2 = r3 = r4 = 1, one sees that (1) is not true for all positive numbers r, r1, r2, r3, r4. This is not surprising, since r is dependent on r1, r2, r3, r4 by the elegant relation

which we shall prove in Theorem 2 below; see also, for example, [2, (5), §266, p. 92], [3. §41, 2°, (1), p. 76] and [4, Problem 6′(i), p. 39]. Using this relation, one can rewrite (1) as

Intuitively, the four numbers r1, r2, r3, r4 are independent, and one may thus ask whether the inequality (3) holds for all positive numbers r1, r2, r3, r4 regardless of being the exradii of some tetrahedron or not.