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A pair of distinct free homotopy classes of closed curves in an orientablesurface $F$ with negative Euler characteristic is said to be length equivalentif for any hyperbolic structure on $F$, the length of the geodesicrepresentative of one class is equal to the length of the geodesicrepresentative of the other class. Suppose $\alpha$ and $\beta$ are twointersecting oriented closed curves on $F$ and $P$ and $Q$ are any twointersection points between them. If the two terms $\langle\alpha*_P\beta\rangle$ and $\langle\alpha*_Q\beta\rangle$ in$[\langle\alpha\rangle,\langle\beta\rangle]$, the Goldman bracket between them,are the same, then we construct infinitely many pairs of length equivalentcurves in $F.$ These pairs correspond to the terms of the Goldman bracketbetween a power of $\alpha$ and $\beta$. As a special case, our constructionshows that given a self-intersecting geodesic $\alpha$ of $F$ and anyself-intersection point $P$ of $\alpha$, we get a sequence of such pairs.Furthermore if $\alpha$ is a filling curve then these pairs are also filling.

Goldman bracket and length equivalent filling curves