I'm solving one hard problem in my homework textbook (it is from the list of hardest problems in the end of book with stars). I reduced it to very simple question which I can prove by two, but very complicated and long ways (using Heron formula and some long algebra operations).
It must be easy and simple (I hope) solution to this question, which i can't see.
Question is: We have two parallel lines $(l_1,l_2)$ and the distance between this lines $|DE|=n$ an integer(see pic.), $n \in \mathbb{N}$. Let points $A$,$B \in l_1$ and $|AB|=|DE|=n$. Let point $C \in l_2$ and $|AC|=k,|BC|=m$. Prove that there are no exist such point $C$ that $k$ and $m$ both integer. (If $k,n \in \mathbb{N}$, then $m \notin \mathbb{N}$ or if $m,n \in \mathbb{N}$, then $k \notin \mathbb{N}$).
I can prove it (like i said before it is very long analysis of equation which we can obtain using formulas for area), I'm looking for simple and short solution. Thanks.
In my proof I use equation
$ 4n^4=(n+k+m)(k+m -n)(n+k-m)(n-(k-m)), \ \text{if } x=k+m, y=k-m \Rightarrow $
$ 5n^4-(x^2+y^2) n^2 +x^2 y^2=0. $