Inverse-square Gaussian law of Brownian first passage (${\tau }_x\stackrel{d}{=}{x}^2/N^2$)

The Brownian reflection principle gives $\mathbb{P}({\tau }_x\le t)=\mathbb{P}(|N|\ge x/\sqrt{t})$ for a standard normal random variable $N$. This is the distribution function of $x^2/N^2$. An independent Brownian motion observed at this hitting time has the Cauchy distribution with scale $x$.