Simplified rational distance problem

Question

① Is there a point on a square with sides of rational length that is a rational distance from each vertex?

Note that this is a very specific case of the Rational Distance Problem, which can be expressed more simply as:

② Are there any Pythagorean triples (L, a, p) and (L, b, q) such that a + b = L?

I've done a computer search for small values (L < 2³¹) and found these pairs of triples such that a + b = L ± 1, but none any closer:

+-----------+----+-----------+-----------+
|     L     |  ± |     a     |     b     |
+-----------+----+-----------+-----------+
|      1344 | -1 |       760 |       583 |
+-----------+----+-----------+-----------+
|      1600 | -1 |      1200 |       399 |
+-----------+----+-----------+-----------+
|      1508 | +1 |      1344 |       165 |
+-----------+----+-----------+-----------+
|     29040 | -1 |     28798 |       241 |
+-----------+----+-----------+-----------+
|    142912 | -1 |     82695 |     60216 |
+-----------+----+-----------+-----------+
|    594220 | -1 |    529371 |     64848 |
+-----------+----+-----------+-----------+
|   3751488 | +1 |   3049984 |    701505 |
+-----------+----+-----------+-----------+
|  17980480 | -1 |  10406328 |   7574151 |
+-----------+----+-----------+-----------+
|  40436800 | -1 |  30470799 |   9966000 |
+-----------+----+-----------+-----------+
|  65395616 | +1 |  34753887 |  30641730 |
+-----------+----+-----------+-----------+
|  95021927 | +1 |  87275664 |   7746264 |
+-----------+----+-----------+-----------+
| 189632508 | +1 | 168503165 |  21129344 |
+-----------+----+-----------+-----------+
| 234121100 | +1 | 208570992 |  25550109 |
+-----------+----+-----------+-----------+
| 283860000 | -1 | 182418750 | 101441249 |
+-----------+----+-----------+-----------+
| 513405900 | -1 | 395722000 | 117683899 |
+-----------+----+-----------+-----------+

Is there a simple disproof of ②?

Through elementary algebra, a precondition for ② is:

③ Are there non-trivial (i.e. distinct, strictly positive) integer solutions for a² - b² = p² - q²?

Which is equivalent to:

④ Are there non-trivial integer solutions for a² + q² = b² + p²?

Of which there are many examples, e.g. Numbers that are Sums of Squares in Several Ways

Answer 1

For 3, you can write it as $L^2=(a+b)(a-b)=(p+q)(p-q)$ so there are solutions for composite $L$. To make $a,b,p,q$ integers requires that $L$ be odd or a multiple of $4$. For example, $12^2=144=72\cdot 2=36\cdot 4=24 \cdot 6=37^2-35^2=20^2-16^2=15^2-9^2$ This doesn't get you 2, however.