from math import *
from numpy import arange
from itertools import combinations

p = 7
q = p // 2

# all points on the circle
pts = [(cos(x), sin(x)) for x in arange(0, 2*pi, 2*pi / p)]
# all segments of the star
segments = list(zip(pts, pts[q:] + pts[:q]))

def intersection(s1, s2):
    x1, y1 = s1[0]
    x2, y2 = s1[1]
    x3, y3 = s2[0]
    x4, y4 = s2[1]
    d = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4)

    if d == 0:
        return None
    return (((x1*y2 - y1*x2) * (x3 - x4) - (x1 - x2) * (x3*y4 - y3*x4)) / d,
            ((x1*y2 - y1*x2) * (y3 - y4) - (y1 - y2) * (x3*y4 - y3*x4)) / d)

# remove duplicates from all intersections of segments
all_pts = list(set(filter(None, [intersection(s[0], s[1]) for s in combinations(segments, 2)])))

def to_polar(x):
    return (sqrt(x[0]**2 + x[1]**2),
            atan2(x[1],  x[0]))

def from_polar(x):
    return (x[0] * cos(x[1]),
            x[0] * sin(x[1]))

all_pts_polar = list(map(to_polar, all_pts))
all_pts_polar.sort(key=lambda x: -x[1])

# start is at alpha=0, so in the middle, need to put it first and rotate
all_pts_polar = all_pts_polar[len(all_pts_polar) // 2:] + all_pts_polar[:len(all_pts_polar) // 2]
all_pts_polar = [(r, alpha + pi / 2) for (r, alpha) in all_pts_polar]

result = list(map(from_polar, all_pts_polar))