  The Cauchy distribution is usually presented as a mathematical curiosity, an
exception to the Law of Large Numbers, or even as an "Evil" distribution in
some introductory courses. It therefore surprised us when Drton and Xiao (2016)
proved the following result for $m=2$ and conjectured it for $m\ge 3$. Let $X=
(X_1,..., X_m)$ and $Y = (Y_1, ...,Y_m)$ be i.i.d $N(0,\Sigma)$, where
$\Sigma=\{\sigma_{ij}\}\ge 0$ is an $m\times m$ and \textit{arbitrary}
covariance matrix with $\sigma_{jj}>0$ for all $1\leq j\leq m$. Then $$Z =
\sum_{j=1}^m w_j \frac{X_j}{Y_j} \ \sim \mathrm{Cauchy}(0,1),$$ as long as
$w=(w_1,..., w_m) $ is independent of $(X, Y)$, $w_j\ge 0, j=1,..., m$, and
$\sum_{j=1}^m w_j=1$. In this note, we present an elementary proof of this
conjecture for any $m \geq 2$ by linking $Z$ to a geometric characterization of
Cauchy(0,1) given in Willams (1969). This general result is essential to the
large sample behavior of Wald tests in many applications such as factor models
and contingency tables. It also leads to other unexpected results such as $$
\sum_{i=1}^m\sum_{j=1}^m \frac{w_iw_j\sigma_{ij}}{X_iX_j} \sim
{\text{L\'{e}vy}}(0, 1). $$ This generalizes the "super Cauchy phenomenon" that
the average of $m$ i.i.d. standard L\'evy variables (i.e., inverse chi-squared
variables with one degree of freedom) has the same distribution as that of a
single standard L\'evy variable multiplied by $m$ (which is obtained by taking
$w_j=1/m$ and $\Sigma$ to be the identity matrix).
