In a paper with the same title, Carlitz [1] proves two inequalities about a triangle ABC and an internal point O:

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show later on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.

Given a triangle with vertices P1, P2, P3 its Fermat point is defined as that point F which minimises the sum P1F + P2F + P3F. It is known that if each of the angles P1, P2, P3 is less than 120°, then F will be the point inside the triangle at which each pair of P1, P2, P3 subtend an angle of 120°, while if any of the angles at P1, P2, P3 exceed 120° then F will coincide with that vertex – see Figure (1).

If the above phrase appeared in an A-level paper the Board responsible would receive many protests: “a contradiction in terms”, “impossible”. But is it? Our attention was drawn to this via this A-level problem from the J.M.B.

(Further Maths, Paper II, 1988)

Population trends are determined by fertility and mortality. The crudest method of measuring fertility and mortality consists in relating the number of births and of deaths to the total population. The birth and death rates thus obtained show the proportion by which a population increases through births and decreases through deaths. But since these rates are calculated without regard to the sex and age composition of the population they do not afford an adequate gauge for the measurement of fertility and mortality The disturbing influence of the sex and age composition is eliminated by computing fertility tables and life tables.