Thank you for your response to my query. Your work on diagonals is spot on. (I view this same development in terms of what I have called "apparent Pythagagorean triples", ie, empirically discovered near triples such as the 70-70-99 you mention. Such triples are empirically all but indistinguishable from "true" triples, and appear to have been used in pyramid design throught the Old Kingdom period.)
The square root of 2 is certainly one aspect that comes into play as one extrapolates the inscribed square phenomenon of Problem 48, but I believe there to be others.
Your specific analysis of RMP 48 appears centered on the area of the circle derivation aspect of this problem. I am hoping to learn if anyone has attempted an analysis that has focused on this problem with a treatment of its various other implications.