  By making use of the greatest common divisor's ($gcd$) properties we can
highlight some connections between playing billiard inside a unit square and
the Fibonacci sequence as well as the Euclidean algorithm. In particular by
defining two maps $\tau$ and $\sigma$ corresponding to translations and
mirroring we are able to rederive Lam\'{e}'s theorem and to equip it with a
geometric interpretation realizing a new way to construct the golden ratio.
Further we discuss distributions of the numbers $p,q\in \mathbb{N}$ with
$gcd(q,p)=1$ and show that these also relate to the Fibonacci sequence.
