  A planar node set $\mathcal X,$ with $|\mathcal X|=\binom{n+2}{2}$ is called
$GC_n$ set if each node possesses fundamental polynomial in form of a product
of $n$ linear factors. We say that a node uses a line $Ax+By+C=0$ if $Ax+By+C$
divides the fundamental polynomial of the node. A line is called $k$-node line
if it passes through exactly $k$-nodes of $\mathcal X.$ At most $n+1$ nodes can
be collinear in $GC_n$ sets and an $(n+1)$-node line is called maximal line.
The Gasca - Maeztu conjecture (1982) states that every $GC_n$ set has a maximal
line. Until now the conjecture has been proved only for the cases $n \le 5.$
Here we adjust and prove a conjecture proposed in the paper - V. Bayramyan, H.
H., Adv Comput Math, 43: 607-626, 2017. Namely, by assuming that the
Gasca-Maeztu conjecture is true, we prove that for any $GC_n$ set $\mathcal X$
and any $k$-node line $\ell$ the following statement holds:
  Either the line $\ell$ is not used at all, or it is used by exactly
$\binom{s}{2}$ nodes of $\mathcal X,$ where $s$ satisfies the condition
$\sigma:=2k-n-1\le s\le k.$ If in addition $\sigma \ge 3$ and $\mu(\mathcal
X)>3$ then the first case here is excluded, i.e., the line $\ell$ is
necessarily a used line. Here $\mu(\mathcal X)$ denotes the number of maximal
lines of $\mathcal X.$
  At the end, we bring a characterization for the usage of $k$-node lines in
$GC_n$ sets when $\sigma=2$ and $\mu(\mathcal X)>3.$
