Given three equivalent statements, prove equality between two sets of variables

Question

What technique should I use to solve the following problem?

Would I utilize the division algorithm?

Let $ m, n, r, s ∈ \mathbf{Z}$.

If $m^2 + n^2 = r^2 + s^2 = mr + ns$,

prove that $m = r$ and $n = s$.

Answer 1

Add $m^2 + n^2$ to $r^2 + s^2$ noting that each of the two sums equals $mr + ns$

$$ m^2 + n^2 + r^2 + s^2 = 2mr + 2ns \\ m^2 - 2mr + r^2 + n^2 - 2ns + s^2 = 0 \\ (m - r)^2 + (n - s)^2 = 0 $$

which for $m, n, r, s$ integer holds if and only if $m = r$ and $n = s$.

Answer 2

We have $$ (m-r)^2+(n-s)^2=m^2-2mr+r^2+n^2-2ns+s^2=0. $$

Hence $m-r=0$ and $n-s=0$.