Orienteering Competition#6355
During the time when Pan Kotsky was competing in a bicycle race, Kotyhoroshko and Vernyhora decided to join an orienteering competition. Starting at the point with coordinates (X_0, Y_0), they received a map and the coordinates of the next point (X_1, Y_1). Upon reaching this point, they were given the coordinates of the second point (X_2, Y_2), and so on, until they reached the last point, the (N-1)-th point.
After reaching the last point, they returned to the starting point and plotted their entire route on the map. Vernyhora remarked, "Look, Kotyhoroshko, our route looks like a regular non-degenerate -gon". "No, it doesn't," replied Kotyhoroshko, "that's not the case". Help the friends resolve this disagreement.
The first line contains an integer (3 \leq N \leq 100) – the number of points on the route. Each of the following lines contains two real numbers X_i and Y_i (0 \leq X_i, Y_i \leq 1), separated by a space, representing the coordinates of the -th point. While some points may have the same coordinates, it is guaranteed that at least one pair of points is at least units apart.
Coordinates are provided with a precision of at least 10^{-10}.
If it is impossible to arrange the points as the vertices of a regular -gon in the order they were visited, print NO on a single line. Otherwise, print YES.
It is guaranteed that if the answer is negative, the coordinates cannot be adjusted by less than 10^{-5} to form the vertices of a regular -gon in the given order.
Example 1