Routhi teoreem

\frac{S_{GHI}}{S_{ABC}}=\frac{(xyz-1)^{2}}{(xy+x+1)(yz+y+1)(zx+z+1)}

X = 7 (slider from 1 to 8, step 1)
Y = 4 (slider from 1 to 8, step 1)
Z = 1 (slider from 1 to 8, step 1)
A = (-1.8254545454545457, 0.3090909090909051)
B = (8.483636363636366, -0.3272727272727315)
C = (3.0109090909090925, 5.581818181818179)
t1, c, a, b = Polygon(A, B, C)
d = Circle(A, c / (X + 1))
e = Circle(B, a / (Y + 1))
f = Circle(C, b / (Z + 1))
D = Intersect(d, c)
E = Intersect(e, a)
F = Intersect(f, b)
g = Segment(A, E)
h = Segment(B, F)
i = Segment(C, D)
G = Intersect(g, i)
H = Intersect(g, h)
I = Intersect(h, i)
t2, i_1, g_1, h_1 = Polygon(G, H, I)
lug: (((X * Y) * Z) - 1)^(2)
nim: ((((X * Y) + X + 1) * ((Y * Z) + Y + 1)) * ((X * Z) + Z + 1))
lugtaand: lug / GCD[lug, nim]
nimtaand: nim / GCD[lug, nim]
text2: "\frac{S_{GHI}}{S_{ABC}}=\frac{(" + (LaTeX[X]) + "·" + (LaTeX[Y]) + "·" + (LaTeX[Z]) + "-1)^{2}}{(" + (LaTeX[X]) + "·" + (LaTeX[Y]) + "+" + (LaTeX[X]) + "+1)·(" + (LaTeX[Y]) + "·" + (LaTeX[Z]) + "+" + (LaTeX[Y]) + "+1)·(" + (LaTeX[Z]) + "·" + (LaTeX[X]) + "+" + (LaTeX[Z]) + "+1)}=\frac{" + (LaTeX[lug]) + "}{" + (LaTeX[nim]) + "}=\frac{" + (LaTeX[lugtaand]) + "}{" + (LaTeX[nimtaand]) + "}
"