Are there integers a, b, c, d generating four right triangles with integer sides?

Question

To make this more precise, we are looking for four (ETA: distinct) positive integers $a$, $b$, $c$, and $d$, such that $\sqrt{a^2+b^2}$, $\sqrt{b^2+c^2}$, $\sqrt{c^2+d^2}$, and $\sqrt{d^2+a^2}$ are all integers as well.

Equivalently, we seek a convex quadrilateral with integer sides, whose diagonals intersect at right angles at a point a (ETA: distinct) integer distance from all four vertices.

ETA: Answered in the affirmative below, by computer search. Is there a more elegant, less brute-forcey way to such an answer?

Answer 1

Computer search finds many examples; considering only those where all four numbers are distinct, we have for example:

$$\begin{align} a & = 6375 \\ b& = 6512 \\ c & = 9984 \\d & = 800 \end{align}$$

and $$\begin{align} a & = 3472 \\ b& = 7296 \\ c & = 10400 \\d & = 2175 \end{align}$$

Answer 2

$$ a=3, b=4, c=3, d=4........... $$