We deal with an extension of the six-point circle theorem for the quadrilateral [1] when the Van Aubel configuration is generalised as in [2] and [3]: similar parallelograms are constructed, all internally or all externally, on the sides of a given quadrilateral.

Martin Josefsson [1] has coined the term ‘bisect-diagonal quadrilateral’ for a quadrilateral with at least one diagonal bisected by the other diagonal, and extensively explored some of its properties. This quadrilateral has also been called a ‘bisecting quadrilateral’ [2], a ‘sloping-kite’ or ‘sliding-kite’ [3], or ‘slant kite’ [4]. The purpose of this paper is to explore some more properties of this quadrilateral.

1. Introduction

In the article [1], a follow-up to my article [2], Christian and Trustrum cite empirical evidence that the probability of a family giving birth to a boy or to a girl may vary from family to family. Letting bi and gi = 1 − bi respectively denote the probability of the ith family giving birth to a boy or to a girl, the suggestion is that bi may be distinct from bj for distinct i and j. As they point out, this has implications for the expected society-wide number of boys and girls. Following my discussion of the ‘stop after k girls’ policy, Christian and Trustrum introduce the related policy ‘stop after k girls or N children’. They argue that the expected number of children under this latter policy is an increasing function of bi (when 0 < k < N). The present article complements their discussion by examining this alternative stopping policy in more detail.

“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1]

Vincenzo Viviani, a 17th century mathematician, proved that the sum of the (perpendicular) distances from a point within an equilateral triangle to its sides is constant as shown in Figure 1. The theorem, named after him, generalises to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel. Viviani's theorem is easily proved by summing the areas of triangles APB, BPC and CPA, equating to the area of the triangle ABC, and then simplifying to obtain h1 + h2 + h3 = H, where H is the altitude of ABC.

The Euler line of a triangle is mostly valued, not for any practical application, but purely as a beautiful, esoteric example of post-Greek geometry. Much to his surprise, however, the author recently came across the following result and theorem by Sylvester (1814-1897) in [1] that involves an interesting application of forces that relate to the Euler line (segment). This result is also mentioned in [2] without proof or reference to Sylvester.