Determine if chasing points will meet

You are given N points in 2D, indexed 0..N-1, forming a cycle.

• At time t = 0, point i is at coordinates (xi, yi).
• All points move simultaneously and continuously.
• Each point moves with constant speed v = 1.
• At any time t, point i moves in the direction of the current position of point (i+1) mod N (i.e., it “chases” the next point).

Task

Write a program that determines whether the points will all meet at the same location at the same time.

If they do meet, output:

1. The meeting time T.
2. The meeting position (X, Y).

If they do not meet, output NEVER.

Input

• Integer N (2 ≤ N ≤ 200)
• N lines of real numbers: xi yi

Output

• Either NEVER, or three real numbers T X Y.

Requirements / Notes

• Your numeric outputs should be within 1e-4 absolute error.
• You may use numerical simulation (time-stepping) as long as it meets the error requirement and runs efficiently.