25725번 - Symmetry: Convex 스페셜 저지다국어

Given is a strictly convex polygon with $n$ vertices $p_1, p_2, \ldots, p_n$ in counterclockwise. Denote $C_i$ as the polygon with $i$ vertices $p_1, p_2, \ldots, p_i$. For each $i=3, 4, \ldots, n$, find the lines which $C_i$ is symmetric about.

There are multiple test cases. The first line of input contains an integer $T$ ($1\le T\le 10^5$), the number of test cases. For each test case:

The first line contains an integer $n$ ($3 \le n \le 3 \cdot 10^5$), the number of vertices.

The $i$-th of the following $n$ lines contains two integers $x_i$, $y_i$ ($-10^9 \le x_i, y_i\le 10^9$), the coordinates of $p_i$.

It is guaranteed that the vertices are given counterclockwise, and the polygon is strictly convex, that is, no three vertices are collinear.

It is guaranteed that the sum of $n$ in all test cases does not exceed $3 \cdot 10^5$.

For each test case:

For each $i=3, 4, \ldots, n$, on the first line, output an integer $k$: the number of lines which $C_i$ is symmetric about.

In each of the following $k$ lines, output three integers $a$, $b$, $c$ ($-2 \cdot 10^{18} \le a, b, c \le 2 \cdot 10^{18}$), denoting that $C_i$ is symmetric about the line $ax+by+c=0$.

If there are multiple answers, you can output any of them. For each $i$, you can output the lines in any order.