2957：Planet Hunting

In a fictitious solar system consisting of star S, a planet P, and its moon M, planet P orbits in a perfect circle around star S with a revolution period of exactly T Earth days, and moon M orbits in a perfect circle around planet P with an unknown revolution period. Given the position of moon M relative to star S at three different time points, your goal is to compute the distance of planet P from star S.

To do this, consider a two-dimensional Cartesian coordinate system with its origin centered at star S. You may assume that P’s counterclockwise orbit around S and M’s counterclockwise orbit around P both lie completely within the x – y coordinate plane. Let (x₁, y₁) denote the position of the moon M on the first observation, let (x₂, y₂) denote its position k₁ Earth days later, and let (x₃, y₃) denote its position k₂ Earth days after the second observation.

The input test file will contain multiple test cases. Each test case consists of two lines. The first line contains the integers T, k₁, and k₂, where 1 ≤ T, k₁, k₂ ≤ 1000. The second line contains six floating-point values x₁, y₁, x₂, y₂, x₃, and y₃. Input points have been selected so as to guarantee a unique solution; the final distance from planet P to star S will always be within 0.1 of the nearest integer. The end-of-file is denoted with a single line containing “0 0 0”.