  An $n$-poised node set $\mathcal X$ in the plane is called $GC_n$ set if the
(bivariate) fundamental polynomial of each node is a product of n linear
factors. A line is called $k$-node line if it passes through exactly $k$-nodes
of $\mathcal X.$ An $(n+1)$-node line is called maximal line. The well-known
conjecture of M. Gasca and J. I. Maeztu states that every $GC_n$ set has a
maximal line. Untill now the conjecture has been proved only for the cases $n
\le 5.$ We say that a node uses a line if the line is a factor in the node's
fundamental polynomial. It is a simple and well-known fact that any maximal
line $M$ is used by all $\binom{n+1}{2}$ nodes in $\mathcal X\setminus M.$ Here
we consider the main result of the paper - V. Bayramyan, H. Hakopian, On a new
property of n-poised and $GC_n$ sets, Adv Comput Math, 43, (2017) 607-626,
stating that any $n$-node line of $GC_n$ set is used either by exactly
$\binom{n}{2}$ nodes or by exactly $\binom{n-1}{2}$ nodes, provided that the
Gasca-Maeztu conjecture is true.
  In this paper we show that this result is not correct in the case $n=3.$
Namely, we bring an example of a $GC_3$ set and a $3$-node line there which is
not used at all. Fortunately, then we were able to establish that this is the
only possible counterexample, i.e., the above mentioned result is true for all
$n\ge 1, n\neq 3.$
  We also characterize the exclusive case $n=3$ and present some new results on
the maximal lines and the usage of $n$-node lines in $GC_n$ sets.

Combinatorics;Numerical Analysis