VWO 2006 Problem 2

Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$ . $Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$ . $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$ . And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$ . $Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$ . $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$ . And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$ . Determine $\frac{|PB|}{|AB|}$ if $S=S'$ .

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