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: Wrong Answer)

import math

EPS = 1e-16

def cross(a, b):
    return a.real * b.imag - a.imag * b.real


def dist(a, b):
    return math.hypot(a.real - b.real, a.imag - b.imag)


def crossPointLL(ln1, ln2):
    u = ln1[1] - ln1[0]
    v = ln2[1] - ln2[0]
    return ln1[0] + u * (cross(v, ln2[0] - ln1[0]) / (EPS + cross(v, u)))


def circumscribed_circle(p1, p2, p3):
    p12 = (p1 + p2) / 2
    p23 = (p2 + p3) / 2
    ln1 = [p12, p12 + (p1 - p2) * complex(0, 1)]
    ln2 = [p23, p23 + (p2 - p3) * complex(0, 1)]
    c = crossPointLL(ln1, ln2)
    return [c, dist(c, p1)]


def inner_product(v1, v2):
    return v1.real * v2.real + v1.imag * v2.imag


def judge_lines_relation(l1_start, l1_end, l2_start, l2_end):
    v1 = l1_end - l1_start
    v2 = l2_end - l2_start
    return abs(cross(v1, v2)) < EPS


from itertools import combinations

N, *XY = map(int, open(0).read().split())
XY = [complex(x, y) for x, y in zip(*[iter(XY)] * 2)]

if N == 2:
    print(dist(XY[0], XY[1]) / 2)
    quit()

ans = float("inf")
for a, b, c in combinations(XY, 3):
    center, r = circumscribed_circle(a, b, c)

    if judge_lines_relation(a, b, b, c) or judge_lines_relation(a, c, c, b) or judge_lines_relation(a, b, a, c):
        r = max(dist(a, b), dist(b, c), dist(c, a))
        for x, y in [(a, b), (b, c), (c, a)]:
            if dist(x, y) == r:
                center = (x + y) / 2

        r /= 2

    if all(dist(xy, center) <= r + EPS for xy in XY):
        ans = min(ans, r)

print("{:.9f}".format(ans))