Volume 42
, Issue 1

2006

Welcome to the first issue of Parabola Incorporating Function for 2006. The first two articles by Mike Hirschhorn are good illustrative examples of the sort of recreational games that you can play with mathematics.

It is well-known that
$$
\log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-+\ \cdots\ \text{for}\ 0\le x\le1,
$$
where the $-+\ \cdots$ notation is used to indicate that successive terms are
alternately subtracted and added.

In this article I am going to assume you are a little bit familiar with the inverse tangent function, $\tan^{-1}x=\text{arctan}{x}.$ There are several properties of this function that you need to know.

The story I want to tell in this issue concerns a development I first learned of a little over 30 years ago, but whose roots lie much deeper than that. It was in 1973 that I was first introduced to the mathematical development known as Catastrophe Theory.

Recently a colleague from the Optometry school came to me with a problem. He had designed a new shaped contact lens. Unlike other lens systems this was not designed as an optical lens, but rather as a shaping lens, designed to be put in overnight.

Q1201 Let $x_1$ and $x_2$ be the solutions of
$$ x^2 - (a+d)x + ad - bc = 0.$$
Prove that $x_1^3$ and $x_2^3$ are the solutions of
$$x^2 - (a^3 + d^3 + 3abc + 3bcd)x + (ad-bc)^3 = 0.$$

Q1191 Let $ABC$ be a triangle with sides $a,b,c$ in the usual way and let $r$ be its circumradius.

  1. Show that $\displaystyle 3r > \frac{a+b+c}{2}.$
  2. Let $P$ be any point within $ABC$, and let $r_1$ be the circumradius of $ABP.$  Is it true that $r_1 < r?$