Bulldoze this triangle

We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$ , distinct points in the plane of $ABC$ such that $PA \cdot BC = PB \cdot CA = PC \cdot AB$ and $QA \cdot BC = QB \cdot CA = QC \cdot AB$ . Let $XY$ be a segment of unit length in a plane $\mathcal{P}$ , and let $\mathcal{S}$ be the region of $\mathcal{P}$ that the bulldozer of $XYZ$ sweeps through as $Z$ varies across the points in $\mathcal{P}$ satisfying $XZ = 2YZ$ . Find the greatest integer that is less than $100$ times the area of $\mathcal{S}$ .