  In this paper we study the geodesic continued fraction in the case of the
Shimura curve coming from the $(2,3,7)$-triangle group. We construct a certain
continued fraction expansion of real numbers using the so-called coding of the
geodesics on the Shimura curve, and prove the Lagrange type periodicity theorem
for the expansion which captures the fundamental relative units of quadratic
extensions of $\mathbb{Q}(\cos(2\pi/7))$ with rank one relative unit groups. We
also discuss the convergence of these continued fractions.
