Find the Triangle

Notice that $\dfrac{56\sqrt{19}}{19} = \dfrac{56}{\sqrt{19}}.$ , so the solution is $56+19=\boxed{75}$

I tried really hard to find an analytical solution, but there were too many independent variables to differentiate. If someone has a suggestion, I would love to hear it.

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# https://brilliant.org/problems/find-the-triangle-2/

from sys import argv
from sympy import *
from itertools import product

# return square of triangle area
def heron(x):
    a, b, c = map(Integer, x)
    s = (a + b + c)/2
    return s*(s - a)*(s - b)*(s - c)

# distance between O and H according to Wolfram
# see eqn (5) on https://mathworld.wolfram.com/Circumcenter.html
def OH(x):
    a, b, c = x
    t2 = a**6 - b**2*a**4 - c**2*a**4 - b**4*a**2 - c**4*a**2 + 3*b**2*c**2*a**2 + b**6 + c**6 - b**2*c**4 - b**4*c**2
    area2 = heron(x)
    OH2 = t2 / (16 * area2)
    return sqrt(OH2)

# triangle inequality
def is_triangle(x):
    a, b, c = x
    return a < b+c and b < a+c and c < a+b

# exhaustively generate all integer triangles
def integer_triangles(n):
    uniq = set()
    sides = list(range(1, n))
    for t in product(sides, repeat=3):
        if is_triangle(t):
            uniq.add(tuple(sorted(t)))
    return list(uniq)

n = int(argv[1]) if len(argv) == 2 else 10
tmax = -1
tri = integer_triangles(n)
for t in tri:
    if OH(t) > tmax:
        tmax = OH(t)
        abc = t
print(abc, tmax)

1

(5, 5, 9) 56*sqrt(19)/19