Width of a Point Set

Consider a set $S$ of $n$ points in the plane. The width $w$ of $S$ is the minimum distance between two parallel lines that enclose $S$. Figure 1 shows an example.

Figure 1: The width $w$ of a set of three points.

There $S$ consists of the $n = 3$ points $(0, 0)$, $(0, 3)$ and $(3, 0)$. The width is realized by the two lines $\ell$ and $\ell'$, whose distance is $w = 3\sqrt{2}/2 \simeq 2.12$. In this task you are given a set of points and you have to compute the integer part of $w^2$ and print it. For Figure 1, $w = 3\sqrt{2}/2$, so $w^2 = 4.5$, and you print the integer part of $4.5$, which is $4$.

Here is a formula that helps. Let $A = (x_a, y_a)$, $B = (x_b, y_b)$ and $C = (x_c, y_c)$ be three points. The height $h$ of the triangle $ABC$ (see Figure 2) is

$h = \sigma \frac{(x_a - x_c)(y_b - y_c) - (x_b - x_c)(y_a - y_c)}{\sqrt{(x_a - x_b)^2 + (y_a - y_b)^2}}$

where $\sigma = 1$ if $ABC$ is counterclockwise (as in Figure 2) and $\sigma = -1$ if $ABC$ is clockwise.

Figure 2: Triangle $ABC$.

If all points of $S$ lie on one straight line, the width $w$ is zero.

The first line contains the integer $n$, the number of points in $S$. Each of the next $n$ lines contains the $x$ coordinate and the $y$ coordinate of one point, separated by a single space.

The coordinates are integers between 0 and 199 inclusive. There are at most 100000 points, and the same point may appear several times.

Print the integer part of $w^2$.