<s>[INST] Geometry - Rectangle ABCD with inside point E. Find the least possible value for sum of interger distances from E to 4 vertices. 
The point E lies within the rectangle ABCD.
If the distances from the vertices to E are all distinct integers, what is the least possible value of AE + BE + CE + DE?
 [/INST] We want
$$\left\{
\begin{align*}
a^2&+c^2&=p^2& \hspace{4em} (1)\\
a^2&+d^2&=q^2 &\hspace{4em} (2)\\
b^2&+c^2&=r^2 &\hspace{4em} (3)\\
b^2&+d^2&=s^2 &\hspace{4em} (4)
\end{align*}
\right.$$
where $a+b$ and $c+d$ are the side lengths of the rectangle, and $p,q,r,s$ are the distances to the vertices.  We want to find a way such that $p,q,r,s$ are distinct integers (i.e. let $\gcd(p,q,r,s)=1$).
\begin{align*}
(1)-(2): && c^2-d^2&=p^2-q^2 &\hspace{4em} (5) \\
(3)-(4): && c^2-d^2&=r^2-s^2 &\hspace{4em} (6) \\
(5)-(6): && 0 & = p^2-q^2-r^2+s^2 \\
&&r^2-s^2 &=p^2-q^2 & \hspace{4em}(7)
\end{align*}
A solution (and the solution containing the smallest numbers) for the diophantine equation $(7)$ is
$$(p,q,r,s)=(8,4,7,1).$$
Therefore, the answer to your question is the sum of $p,q,r,s$, or $20$.
More info:  https://oeis.org/A118882
</s>