Using Heron's formula for area, calculate the sum of the area of the three small triangles and compare that to the area of the large trianle.  They should be equal.  Adjust the value of 'r' until the difference is very small.

I found r = about  1.07006369426752

Python code follows:

def areaHeron(a,b,c):
    """ input 3 float numbers, the 3 sides of a triangle
    output the area """
    s = (a+b+c)/2  # semiperimeter
    return (s*(s-a)*(s-b)*(s-c))**(1/2)

def calcR(r):
    """ input radius guess for the small circle,
    return the error between the calculated area of
    the three smaller triangles vs the total triangle
    specifically for radii of 6, 7, 8 units """
    a1 = areaHeron(6+7,6+r,7+r)
    a2 = areaHeron(6+8,6+r,8+r)
    a3 = areaHeron(7+8,7+r,8+r)
    aTotal = areaHeron(6+7,6+8,7+8)
    return a1+a2+a3 - aTotal
   
epsilon = .0000000000001
hi = 2  # initial guesses for r and the range of r
lo = 1
r= 1.5
while abs(calcR(r)) > epsilon:    # divide and conquer search for best r
    if calcR(r) > 0:
        hi = r
        r = (lo + r)/2
    elif calcR(r) < 0:
        lo = r
        r = (r + hi)/2
    elif  calcR(r) == 0:
        print('found r exactly')
        break
print(r, calcR(r))

Output:  1.0700636942675175 5.684341886080802e-14