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

Geodesic continued fraction for Shimura curves and its periodicity: the case of $(2,3,7)$-triangle group