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?
