Let $\mathscr{L}(S^1)$ be the Lagrange spectrum arising from intrinsic Diophantine approximation of a unit circle $S^1$ by its rational points. We give a complete description of the structure of $\mathscr{L}(S^1)$ below its smallest accumulation point... We use digit expansions of points on $S^1$, which was originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $\mathscr{L}(S^1)$ is 2. Also we characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of Romik's digit expansions. Our theorem is an analogue of a celebrated theorem of Markoff on badly approximable real numbers. (read more)