import sympy
import joblib
import tqdm

def gcd(a, b):
    if b == 0: return a
    return gcd(b, a % b)

class Point:
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __str__(self):
        return f"({self.x}, {self.y})"

    def magnitude(self):
        return self.x * self.x + self.y * self.y

    def distance_to(self, other):
        dx = other.x - self.x
        dy = other.y - self.y
        return sympy.sqrt(dx * dx + dy * dy)

def circumcenter(A: Point, B: Point, C: Point) -> Point:
    D = sympy.S(2) * (A.x * (B.y - C.y) + B.x * (C.y - A.y) + C.x * (A.y - B.y))
    Ux = (A.magnitude() * (B.y - C.y) + B.magnitude() * (C.y - A.y) + C.magnitude() * (A.y - B.y)) / D
    Uy = (A.magnitude() * (C.x - B.x) + B.magnitude() * (A.x - C.x) + C.magnitude() * (B.x - A.x)) / D
    return Point(Ux, Uy)

def lerp(src: Point, dst: Point, dist: float) -> Point:
    total = src.distance_to(dst)
    ratio = sympy.Rational(dist, total)
    dx = src.x + (dst.x - src.x) * ratio
    dy = src.y + (dst.y - src.y) * ratio
    return Point(dx, dy)

def heron(a, b, c):
    a, b, c = sympy.S(a), sympy.S(b), sympy.S(c)
    p = (a + b + c) / sympy.S(2)
    return sympy.sqrt(p * (p - a) * (p - b) * (p - c))

def inv_law_cosines(a, b, c):
    # given a, b, c return C
    return sympy.acos((a * a + b * b - c * c) / (sympy.S(2) * a * b))


def compute_d(A: Point, B: Point, C: Point, r1: float, r2: float, r3: float) -> float:
    r1, r2, r3 = sympy.S(r1), sympy.S(r2), sympy.S(r3)

    ABmid = lerp(A, B, r1)
    BCmid = lerp(B, C, r2)
    CAmid = lerp(C, A, r3)
    D = circumcenter(ABmid, BCmid, CAmid)

    re = sympy.symbols("re")
    t1 = heron(r1 + r3, r1 + re, r3 + re)
    t2 = heron(r1 + r2, r1 + re, r2 + re)
    t3 = heron(r2 + r3, r2 + re, r3 + re)
    tfull = heron(r1 + r2, r1 + r3, r2 + r3)
    eq = sympy.Eq(t1 + t2 + t3, tfull)
    result = sympy.solve(eq, re)
    result = result[0].evalf()
    # print(f"radius of E is {result}")

    EAB = inv_law_cosines(r1 + result, r1 + r2, r2 + result)
    Ex = (r1 + result) * sympy.cos(EAB)
    Ey = (r1 + result) * sympy.sin(EAB)
    E = Point(Ex, Ey)

    # print(D, E)
    return D.distance_to(E).evalf()

total = 0
count = 0
def create_loop():
    for r1 in range(1, 101 - 2):
        for r2 in range(r1 + 1, 101 - 1):
            for r3 in range(r2 + 1, 101):
                if gcd(gcd(r1, r2), r3) != 1: continue
                yield (r1, r2, r3)

def calc(x):
    r1, r2, r3 = x
    # build a triangle out of the side lengths
    # A is at (0, 0), a = r2 + r3
    # B is at (c, 0), b = r1 + r3
    # c = r1 + r2
    # C is at ()
    a = sympy.S(r2 + r3)
    b = sympy.S(r1 + r3)
    c = sympy.S(r1 + r2)

    A = Point(sympy.S(0), sympy.S(0))
    B = Point(c, sympy.S(0))
    
    CAB = sympy.acos(sympy.Rational(b * b + c * c - a * a, 2.0 * b * c))
    Cx = b * sympy.cos(CAB)
    Cy = b * sympy.sin(CAB)
    C = Point(Cx, Cy)

    d = compute_d(A, B, C, r1, r2, r3)
    return d
    print(f"d({r1}, {r2}, {r3}) = {d}")

total = len(list(create_loop()))
L = joblib.Parallel(n_jobs=8)(joblib.delayed(calc)(tup) for tup in tqdm.tqdm(create_loop(), total=total))
print(sum(L) / len(L))