D - Congruence Points

Score : $400$ points

Problem Statement

You are given two sets $S=\{(a_1,b_1),(a_2,b_2),\ldots,(a_N,b_N)\}$ and $T=\{(c_1,d_1),(c_2,d_2),\ldots,(c_N,d_N)\}$ of $N$ points each on a two-dimensional plane.

Determine whether it is possible to do the following operations on $S$ any number of times (possibly zero) in any order so that $S$ matches $T$.

• Choose a real number $p\ (0 \lt p \lt 360)$ and rotate every point in $S$ $p$ degrees clockwise about the origin.
• Choose real numbers $q$ and $r$ and move every point in $S$ by $q$ in the $x$-direction and by $r$ in the $y$-direction. Here, $q$ and $r$ can be any real numbers, be it positive, negative, or zero.

Constraints

• $1 \leq N \leq 100$
• $-10 \leq a_i,b_i,c_i,d_i \leq 10$
• $(a_i,b_i) \neq (a_j,b_j)$ if $i \neq j$.
• $(c_i,d_i) \neq (c_j,d_j)$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$a_1$ $b_1$

$a_2$ $b_2$

$\hspace{0.6cm}\vdots$

$a_N$ $b_N$

$c_1$ $d_1$

$c_2$ $d_2$

$\hspace{0.6cm}\vdots$

$c_N$ $d_N$

Output

If we can match $S$ with $T$, print Yes; otherwise, print No.

3
0 0
0 1
1 0
2 0
3 0
3 1

Yes

The figure below shows the given sets of points, where the points in $S$ and $T$ are painted red and green, respectively:

In this case, we can match $S$ with $T$ as follows:

1. Rotate every point in $S$ $270$ degrees clockwise about the origin.
2. Move every point in $S$ by $3$ in the $x$-direction and by $0$ in the $y$-direction.

3
1 0
1 1
3 0
-1 0
-1 1
-3 0

No

The figure below shows the given sets of points:

Although $S$ and $T$ are symmetric about the $y$-axis, we cannot match $S$ with $T$ by rotations and translations as stated in Problem Statement.

4
0 0
2 9
10 -2
-6 -7
0 0
2 9
10 -2
-6 -7

Yes

6
10 5
-9 3
1 -5
-6 -5
6 9
-9 0
-7 -10
-10 -5
5 4
9 0
0 -10
-10 -2

Yes