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 Python (3.4.3) (judge verdict: Wrong Answer)


import numpy as np
import itertools

def outer_soul(dots):
    ra,rb,rc = [np.array(i) for i in dots]
    A = np.dot(rb-rc,rb-rc)
    B = np.dot(rc-ra,rc-ra)
    C = np.dot(ra-rb,ra-rb)

    T = A*(B+C-A)
    U = B*(C+A-B)
    W = C*(A+B-C)

    return (T*ra + U*rb + W*rc)/(T + U + W),ra
    
        
n = int(input())
l = [[int(i) for i in input().split()] for _ in range(n)]

dist = 0
 
for i in range(n):
    for j in range(i+1,n):
        dist = max(dist,abs(l[i][0] - l[j][0]) ** 2 + abs(l[i][1] - l[j][1])**2)
    
ans1 = np.sqrt(dist) / 2 
# 外心の場合
ans2 = 0
if len(l) > 2:
    comb = itertools.combinations(l,3)
    for i in comb:
        print(i)
        center,dot = outer_soul(i)
        print(center,dot)
        ans2 = min(ans2,np.sqrt(((center - dot) **2).sum()))
        print(ans2)
    
print(min(ans1,ans2) if ans2 > 0 else ans1)