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Gauss' classical reduction theory for indefinite binary quadratic forms over$\mathbb{Z}$ has originally been proven by means of purely algebraic andarithmetic considerations. It was later discovered that this reduction theoryis closely related to a certain symbolic dynamics for the geodesic flow on themodular surface, and hence can also be deduced geometrically. In this article,we use certain symbolic dynamics for the geodesic flow on Hecke trianglesurfaces (also the non-arithmetic ones) to develop reduction theories for theindefinite binary quadratic forms associated to Hecke triangle groups.Moreover, we propose an algorithm to decide for any $g\in {\rmPSL}_2(\mathbb{R})$ whether or not $g$ is contained in the Hecke triangle groupunder consideration, and provide an upper estimate for its run time.

A geometric reduction theory for indefinite binary quadratic forms over $\mathbb{Z}[\lambda]$