Diophantic Spider and Fly

Given a rectangular room with three integer sides A<=B<=C, A+B+C=S,
is it possible that all three geodesic distances between extreme corners
are integers as well ?

The answer is positive, and the three minimal solutions are for :

A. S =  833 = 7 *  7 * 17 = 108 + 357 +  368
B. S = 2737 = 7 * 17 * 23 = 564 + 748 + 1425
C. S = 3703 = 7 * 23 * 23 = 348 + 975 + 2380

As is shown by :

A. 108^2 + 725^2 = 733^2
357^2 + 476^2 = 595^2
368^2 + 465^2 = 593^2 (min)

B.  564^2 + 2173^2 = 2245^2
748^2 + 1989^2 = 2125^2
1425^2 + 1312^2 = 1937^2 (min)

C.  348^2 + 3355^2 = 3373^2
748^2 + 2728^2 = 2896^2
2380^2 + 1323^2 = 2723^2 (min)

According to the schematic unfolding as demonstrated
for the case S = 7 = 1 + 2 + 4  [shown below]

The above solutions were arrived at by computational methods,
starting from an "open" list of primitive pythagorian triples
ordered by rising values of both their generating parameters.
Only if S is a composite number there exists several
pythagorian equations and by combining, for given S, any two
of them, a whole set of "near-solutions" is received, that
is : two of the three critical distances  are integers,
while the third in general is not. But a sufficient
condition can easily be given under which the third distance
will be integer as well. This condition is fulfilled sometimes
for bigger values of S as well, but so far all these solutions
proved to be only whole multiplies of the first three, and so
really do not add anything new.

On the other hand, purely statistical considerations show that
using a more composite number for S considerably enlarges the
set of equations, however still more S itself, so that the
chances the sufficient condition to be fulfilled are decreasing.

So, maybe, no other positive solutions exist. But contrary to
this fact, reality may be otherwise. And besides, there well
may exist solution which dont fulfill the sufficient, but not
necessary, condition. All this was done with the aid of an H.P.
scientific pocket-calculator, and may be put on a broader base
by a program on a more adapted computer. Or maybe the problem
can be solved, or was even treated before, by classical methods ?

Scanned shcematic, drawn by Herman Baer