= Solution
For a coordinate curve $v=v_0$, the length between $u_1$ and $u_2$ is
$$
L_v(u_1,u_2)=\int_{u_1}^{u_2}\sqrt{E(u,v_0)}\,du,
$$
where $E$ is a coefficient of the <first fundamental form>. Thus $\partial E/\partial v=0$ makes the two $u$-directed sides of every coordinate rectangle equally long. Similarly,
$$
L_u(v_1,v_2)=\int_{v_1}^{v_2}\sqrt{G(u_0,v)}\,dv,
$$
so $\partial G/\partial u=0$ makes the two $v$-directed sides equally long.
Conversely, suppose every pair of opposite sides has equal length. Fix $v_0,v_1$. Equality of the first pair for every interval $[u_1,u_2]$ gives
$$
\int_{u_1}^{u_2}\left(\sqrt{E(u,v_0)}-\sqrt{E(u,v_1)}\right)du=0.
$$
The <fundamental theorem of calculus> implies that the continuous integrand vanishes, so $E$ is independent of $v$. The same argument shows that $G$ is independent of $u$. Hence the stated equal-length property is equivalent to \b[$E_v=G_u=0$].
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