Four, (Full marks 20 points) As shown in the figure, the side length of the equilateral $\triangle ABC$ is $1$, and points $R, P, Q$ on $AB, BC, CA$ satisfy $AR + BP + CQ = 1$ while moving. Let $BP = x, CQ = y$, $AR = z$, and the area of $\triangle PQR$ be $S$. Try to express $S$ in terms of $x, y, z$.

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\begin{array}{l}\text { RT }, S_{\triangle A B C}=\frac{\sqrt{3}}{4}, S_{\triangle K Q Q}=-\frac{1}{2}(1-y) z \sin 60^{\circ} \\ =\frac{\sqrt{3}}{4}-(1-y) z \text {. } \\ \text { Similarly, } S_{\triangle B P R}=\frac{\sqrt{3}}{4}(1-z) x \text {, } \\ S_{\triangle G Q P}=\frac{\sqrt{3}}{4}(1-x) y \text {. } \\ \text { Therefore, } S=\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{4}[(1-y) z+(1-z) x \\ +(1-x) y] \\ =\frac{\sqrt{3}}{4}(x y+y z+z x) \text {. } \\\end{array}