<p><b>Abstract</b>—Quorum-based mutual exclusion algorithms are resilient to node and communication line failures. Recently, some mutual exclusion algorithms successfully use logical structures to construct coteries with small quorums sizes. In this paper, we introduce a geometric approach on dealing with the logical structures and present some useful geometric properties for constructing coteries and <it>k</it>-coteries. Based on those geometric properties, a logical structure named <it>three-sided graph</it> is proposed to provide a new scheme for constructing coteries with small quorums: The smallest quorum size is <tmath>$O(\sqrt N)$</tmath> in a homogeneous system with <it>N</it> nodes and <it>O</it>(1) in a heterogeneous system. In addition, we also extend the three-sided graph to the <it>n-sided graph</it> for constructing <it>k</it>-coteries.</p>