Joel's Problem 4: Maximum of maximums

Geometry Level pending

Consider a rectangle R with vertices A, B, C, D in clockwise order. Among all points P on segment CD, let the maximum value of \(PA \cdot PB\) be \(f (R)\). Among all rectangles Q of area \(\sqrt {2}+1\), let \(M\) be the maximum value of \(f (Q)\). However, only some rectangles can have a point on segment CD such that \(PA \cdot PB=M\). Let \(S\) be the set of all such rectangles. Find the maximum possible value of \(\frac {AB}{BC}\) over all rectangles in \(S\).

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