I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle. 

## Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does there exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that:
$$
\begin{cases}
\|(t_1,W_{t_1})-(t_2,W_{t_2})\| = \ell\\
\|(t_3,W_{t_3})-(t_2,W_{t_2})\| = \ell\\
\|(t_1,W_{t_1})-(t_3,W_{t_3})\| = \ell\\
\end{cases}
~~~~\mathrm{?}
$$ 

## Illustration

The problem I mention is then to find a equilateral triangle of length $\ell$ as in the drawing.
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/pJAtrAfg.png