Christopher J. Bradley

The Absolute Correlation Coefficient

The two most common measures of central tendency and dispersion in statistics are the mean and standard deviation on the one hand, and the median and absolute deviation on the other. For most purposes the former measure is preferred for two very good reasons; the first is that the squares of quantities are easier to handle analytically than their moduli; and secondly for all the common symmetrical distributions, such as the normal, uniform and binomial distributions, for which the mean and median coincide, if a sample is taken to estimate the central value, then the mean of that sample has a smaller variance than the median, and is therefore relatively more efficient as an estimator of the central value of the parent population.

Hagge Circles and Isogonal Conjugation

Let Q be a point on the circumcircle of triangle ABC . The reflections of Q in the three triangle sides are known to be collinear, and the line thus defined contains the orthocentre H . This fact can form the basis of a proof of the existence of the Simson line, or alternatively can be deduced from the existence of the Simson line by enlarging the Simson line by a factor of 2 from the centre Q .

Hexagons with opposite sides parallel

This paper presents a number of theorems about hexagons whose three pairs of opposite sides are parallel. The first of these is a well-known result that the vertices of such a hexagon lie on a conic. Theorems 3 and 4 show how such conies are related to the Cevians of a triangle, and which Cevians lead to such conies being circles. When they are circles they are called Tucker circles. None of the results is at all obvious, yet it seems that some of the results presented here were known in the late nineteenth century or the early twentieth century. They seem to be of more than passing interest which should not get lost by the passage of time.

From integer Lorentz transformations to Pythagoras

In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x 2 + y 2 - z 2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y . These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z , thereby accounting for all primitive triples. There is a bonus in extending the method beyond the first objective in that the theory, which we outline, of the Lorentz transformations involved not only provides necessary background explanation of the first part, but it also enables us to show how to solve by recurrence Diophantine equations of the form x 2 + y 2 = z 2 + n , where n is any integer.