= Solution
Let $d_n=x_{n+1}-x_n$ be the direction vector of the segment $X_nX_{n+1}$. Consecutive vectors $x_n,x_{n+1}$ are perpendicular and $|x_{n+1}|=\lambda|x_n|$. Using $x_{n+2}=-\lambda^2x_n$,
$$
d_n\mathbin{\cdot}d_{n+1}
=(x_{n+1}-x_n)\mathbin{\cdot}(x_{n+2}-x_{n+1})
=-|x_{n+1}|^2+\lambda^2|x_n|^2=0.
$$
Thus adjacent segments are perpendicular. Also
$$
d_{n+2}=x_{n+3}-x_{n+2}=-\lambda^2(x_{n+1}-x_n)=-\lambda^2d_n,
$$
so
$$
\boxed{X_nX_{n+1}\parallel X_{n+2}X_{n+3}}.
$$
If $0<\lambda<1$, the norm formula in part (i) gives $x_n\to0$. If $\lambda=1$, then $x_{n+2}=-x_n$: the points $X_1,X_2,-X_1,-X_2$ repeat cyclically and form a square centered at the origin.
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