The square RSTV is inscribed in the square WXYZ such that R lies on WX, S lies on XY, T lies on YZ and V lies on ZW. There are infinitely many angles by which to rotate RSTV inside WXYZ, changing the size of WXYZ as we go, such that each of the lines WR, RX, XS, SY, YT, TZ, ZV, VW, RS, ST, TV and VR has integer side length; these are the pythagorean triples.
Allow for RSTV to instead be any rectangle with integer side lengths, still touching the square WXYZ as described on at least three of R, S, T and V. Does every ratio of RSTV side lengths have infinitely many angles by which RSTV is rotated where all drawn lines have integer lengths?
If so, in the cases where you have only three of R, S, T and V lying on WXYZ, are there infinitely many cases where the line perpendicular from whichever of WX, XY, YZ and ZW is not touched by RSTV to whichever of R, S, T and V is not touching WXYZ is also an integer length?