Suppose $w$ is a world in an Euclidean frame, and $w\mathbin R v$, then by the Euclidean property $w$ reaches both of the "two" worlds $v$ and $v$ (indeed, the same world), and thus $v\mathbin R v$. So in a Euclidean frame, any world that is reachable from some other world, reaches itself (or every world with an incoming arrow has a reflexive arrow).
Now, suppose $w\vDash\Diamond\Diamond\Diamond\phi$, then we know that $w\mathbin R v$ for some world $v$ such that $v\vDash \Diamond\Diamond\phi$. This, in turn, means that $v\mathrel R u$ for some world $u$ such that $u\vDash \Diamond\phi$. By the above argument we see that $v\mathbin R v$, since $v$ is reachable from the world $w$. Thus we have $v\mathrel R v$ and $v\mathrel R u$, which implies that $u\mathrel R v$ by the Euclidean property. Furthermore, we have $u\mathrel R u$ as well, since $u$ is reachable from $v$.
The final step is to see that because $u\vDash \Diamond \phi$, there is some $z$ such that $u\mathrel R z$ and $z\vDash\phi$. By the same reasoning as before, we can find out that not only $u\mathrel R z$, but also $v\mathrel R z$. This gives us that $w\mathrel R v\mathrel R z$, and thus $v\vDash \Diamond\Diamond\phi$.