Problem:
Score : 600 points

Problem Statement

Given are N points (x_i, y_i) in a two-dimensional plane.

Find the minimum radius of a circle such that all the points are inside or on it.

Constraints

2 \leq N \leq 50

0 \leq x_i \leq 1000

0 \leq y_i \leq 1000

The given N points are all different.

The values in input are all integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the minimum radius of a circle such that all the N points are inside or on it.

Your output will be considered correct if the absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

2
0 0
1 0

Sample Output 1

0.500000000000000000

Both points are contained in the circle centered at (0.5,0) with a radius of 0.5.

Sample Input 2

3
0 0
0 1
1 0

Sample Output 2

0.707106781186497524

Sample Input 3

10
10 9
5 9
2 0
0 0
2 7
3 3
2 5
10 0
3 7
1 9

Sample Output 3

6.726812023536805158

If the absolute or relative error from our answer is at most 10^{-6}, the output will be considered correct.

Solution in Ruby (2.3.3) (judge verdict: Accepted)

require "matrix"

N = gets.to_i
V = []
N.times{
  V << gets.chomp.split(" ").map(&:to_i)
}

def get_center_radius(a, b, c)
  k = Matrix[[a[0] - b[0], a[1] - b[1]], [b[0] - c[0], b[1] - c[1]]]
  d = Matrix[[a[0]**2 - b[0]**2 + a[1]**2 - b[1]**2], [b[0]**2 - c[0]**2 + b[1]**2 - c[1]**2]]
  if k.determinant == 0 then
    [[0, 0], 0]
  else
    c = (k.inverse * d) / 2
    r = c - Matrix[[a[0]], [a[1]]]
    r2 = (r.transpose * r).to_a
    wk = c.to_a
    [[wk[0][0], wk[1][0]], r2[0][0]]
  end
end

radius_squared_max = 0
center = []
V.combination(2){|a, b|
  tmp = ((a[0] - b[0]) ** 2 + (a[1] - b[1]) ** 2) / 4.0
  if radius_squared_max < tmp then
    radius_squared_max = tmp
    center = [(a[0]+b[0])/2.0, (a[1]+b[1])/2.0]
  end
}
if V.all?{|a, b| (a - center[0])**2 + (b - center[1])**2 <= radius_squared_max} then
  puts Math.sqrt(radius_squared_max)
else
  radius3 = 0
  V.combination(3){|a, b, c|
    cent, rad = get_center_radius(a, b, c)
    next if rad <= radius_squared_max
    next if radius3 != 0 && radius3 <= rad
    next if V.any?{|a, b| (a - cent[0])**2 + (b - cent[1])**2 > rad}
    radius3 = rad
  }
  puts Math.sqrt(radius3)
end