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 PyPy3 (2.4.0) (judge verdict: Accepted)

import itertools
import math
N = int(input())
p = []
for i in range(N):
    x, y = map(int, input().split())
    p.append([x, y])
ans = 10000
if len(p) >= 3:
    S = list(itertools.combinations(p,3))
    for X in S:
        ok = True
        a = X[0][0]
        b = X[0][1]
        c = X[1][0]
        d = X[1][1]
        e = X[2][0]
        f = X[2][1]
        if (a-c)*(b-f) == (a-e)*(b-d):
            continue
        py = ((e-a)*(a**2+b**2-c**2-d**2)-(c-a)*(a**2+b**2-e**2-f**2))/(2*(e-a)*(b-d)-2*(c-a)*(b-f))
        if c == a:
            px = (2*(b-f)*py-(a**2+b**2-e**2-f**2))/(2*(e-a))
        else:
            px = (2*(b-d)*py-(a**2+b**2-c**2-d**2))/(2*(c-a))
        r = math.sqrt((px-a)**2+(py-b)**2)
        for Y in p:
            if not (r+0.000001)**2 >= (px-Y[0])**2 + (py-Y[1])**2:
                ok = False
        if ok:
            ans = min(ans, r)
T = list(itertools.combinations(p,2))
for X in T:
    a = X[0][0]
    b = X[0][1]
    c = X[1][0]
    d = X[1][1]
    ok = True
    px = (a+c)/2
    py = (b+d)/2
    r = math.sqrt((px-a)**2+(py-b)**2)
    for Y in p:
        if not (r+0.000001)**2 >= (px-Y[0])**2 + (py-Y[1])**2:
            ok = False
    if ok:
        ans = min(ans, r)
print(ans)