We introduce a new heuristic for the A* algorithm that references a data structure significantly smaller than that of ALT. We characterize the behavior of this new heuristic based on a dual landmark configuration that leverages quadrilateral inequalities to identify the lower bound for shortest path. Using this approach, we demonstrate both the utility and detriments of using polygon inequalities aside from the triangle inequality to establish lower bounds for shortest path queries. While this new heuristic does not dominate previous heuristics based on triangle inequalities, the inverse is true, as well. Further, we demonstrate that an A* heuristic function does not necessarily outperform another heuristic that it dominates. In comparison to other landmark methods, the new heuristic maintains a larger average search space while commonly decreasing the number of computed arithmetic operations. The new heuristic can significantly outperform previous methods, particularly in graphs with larger path lengths. The characterization of the use of these inequalities for bounding offers insight into its applications in other theoretical spaces.