  We consider a planar Brownian motion starting from $O$ at time $t=0$ and
stopped at $t=1$ and a set $F= \{OI_i ; i=1,2,..., n\}$ of $n$ semi-infinite
straight lines emanating from $O$. Denoting by $g$ the last time when $F$ is
reached by the Brownian motion, we compute the probability law of $g$. In
particular, we show that, for a symmetric $F$ and even $n$ values, this law can
be expressed as a sum of $\arcsin $ or $(\arcsin)^2 $ functions. The original
result of Levy is recovered as the special case $n=2$. A relation with the
problem of reaction-diffusion of a set of three particles in one dimension is
discussed.
