In the quadrilateral in question NGN*K :

Let D be the intersection of the lines NG, N*K, resp. and E the intersection of the lines N*G, NK, resp.
Then the circumcircles of DNK, DN*G, EN*K, ENG are concurrent at a point M.
This is the Miquel point of the NGN*K. 
The Simson line of X(23096) wrt triangle ABC is perpendicular to UV, where U is the midpoint of GK and V is the midpoint of NgN.

A = (11.432377256889447, 10.701976560025328)
B = (-2, -4)
C = (9.907195121951215, -3.7658726639214666)
a = Segment(B, C)
b = Segment(C, A)
c = Segment(A, B)
k = Line(A, B)
i = Line(B, C)
j = Line(C, A)
x2 = TriangleCenter(A, B, C, 2)
x5 = TriangleCenter(A, B, C, 5)
x6 = TriangleCenter(A, B, C, 6)
x54 = TriangleCenter(A, B, C, 54)
f = Segment(x2, x6)
U = Midpoint(x2, x6)
g = Segment(x54, x5)
V = Midpoint(x54, x5)
h = Line(U, V)
l = Line(x6, x5)
m = Line(x5, x2)
n = Line(x54, x2)
p = Line(x54, x6)
E = Intersect(n, l)
D = Intersect(m, p)
d = Circle(D, x5, x6)
e = Circle(D, x54, x2)
q = Circle(x6, E, x54)
r = Circle(E, x5, x2)
s = Circle(B, C, A)
x23096 = Barycenter({A, B, C}, {((((a^(2) * (a^(2) + b^(2) - c^(2))) * (a^(2) - b^(2) + c^(2))) * (a^(10) - ((4 * a^(8)) * b^(2)) + ((4 * a^(6)) * b^(4)) + ((2 * a^(4)) * b^(6)) - ((5 * a^(2)) * b^(8)) + (2 * b^(10)) - ((3 * a^(8)) * c^(2)) + (((11 * a^(6)) * b^(2)) * c^(2)) - (((7 * a^(4)) * b^(4)) * c^(2)) + (((10 * a^(2)) * b^(6)) * c^(2)) - ((5 * b^(8)) * c^(2)) + ((2 * a^(6)) * c^(4)) - (((14 * a^(4)) * b^(2)) * c^(4)) - (((7 * a^(2)) * b^(4)) * c^(4)) + ((2 * b^(6)) * c^(4)) + ((2 * a^(4)) * c^(6)) + (((11 * a^(2)) * b^(2)) * c^(6)) + ((4 * b^(4)) * c^(6)) - ((3 * a^(2)) * c^(8)) - ((4 * b^(2)) * c^(8)) + c^(10))) * (a^(10) - ((3 * a^(8)) * b^(2)) + ((2 * a^(6)) * b^(4)) + ((2 * a^(4)) * b^(6)) - ((3 * a^(2)) * b^(8)) + b^(10) - ((4 * a^(8)) * c^(2)) + (((11 * a^(6)) * b^(2)) * c^(2)) - (((14 * a^(4)) * b^(4)) * c^(2)) + (((11 * a^(2)) * b^(6)) * c^(2)) - ((4 * b^(8)) * c^(2)) + ((4 * a^(6)) * c^(4)) - (((7 * a^(4)) * b^(2)) * c^(4)) - (((7 * a^(2)) * b^(4)) * c^(4)) + ((4 * b^(6)) * c^(4)) + ((2 * a^(4)) * c^(6)) + (((10 * a^(2)) * b^(2)) * c^(6)) + ((2 * b^(4)) * c^(6)) - ((5 * a^(2)) * c^(8)) - ((5 * b^(2)) * c^(8)) + (2 * c^(10)))), ((((b^(2) * (a^(2) + b^(2) - c^(2))) * ((-a^(2)) + b^(2) + c^(2))) * ((2 * a^(10)) - ((5 * a^(8)) * b^(2)) + ((2 * a^(6)) * b^(4)) + ((4 * a^(4)) * b^(6)) - ((4 * a^(2)) * b^(8)) + b^(10) - ((5 * a^(8)) * c^(2)) + (((10 * a^(6)) * b^(2)) * c^(2)) - (((7 * a^(4)) * b^(4)) * c^(2)) + (((11 * a^(2)) * b^(6)) * c^(2)) - ((3 * b^(8)) * c^(2)) + ((2 * a^(6)) * c^(4)) - (((7 * a^(4)) * b^(2)) * c^(4)) - (((14 * a^(2)) * b^(4)) * c^(4)) + ((2 * b^(6)) * c^(4)) + ((4 * a^(4)) * c^(6)) + (((11 * a^(2)) * b^(2)) * c^(6)) + ((2 * b^(4)) * c^(6)) - ((4 * a^(2)) * c^(8)) - ((3 * b^(2)) * c^(8)) + c^(10))) * (a^(10) - ((3 * a^(8)) * b^(2)) + ((2 * a^(6)) * b^(4)) + ((2 * a^(4)) * b^(6)) - ((3 * a^(2)) * b^(8)) + b^(10) - ((4 * a^(8)) * c^(2)) + (((11 * a^(6)) * b^(2)) * c^(2)) - (((14 * a^(4)) * b^(4)) * c^(2)) + (((11 * a^(2)) * b^(6)) * c^(2)) - ((4 * b^(8)) * c^(2)) + ((4 * a^(6)) * c^(4)) - (((7 * a^(4)) * b^(2)) * c^(4)) - (((7 * a^(2)) * b^(4)) * c^(4)) + ((4 * b^(6)) * c^(4)) + ((2 * a^(4)) * c^(6)) + (((10 * a^(2)) * b^(2)) * c^(6)) + ((2 * b^(4)) * c^(6)) - ((5 * a^(2)) * c^(8)) - ((5 * b^(2)) * c^(8)) + (2 * c^(10)))), ((((c^(2) * (a^(2) - b^(2) + c^(2))) * ((-a^(2)) + b^(2) + c^(2))) * (a^(10) - ((4 * a^(8)) * b^(2)) + ((4 * a^(6)) * b^(4)) + ((2 * a^(4)) * b^(6)) - ((5 * a^(2)) * b^(8)) + (2 * b^(10)) - ((3 * a^(8)) * c^(2)) + (((11 * a^(6)) * b^(2)) * c^(2)) - (((7 * a^(4)) * b^(4)) * c^(2)) + (((10 * a^(2)) * b^(6)) * c^(2)) - ((5 * b^(8)) * c^(2)) + ((2 * a^(6)) * c^(4)) - (((14 * a^(4)) * b^(2)) * c^(4)) - (((7 * a^(2)) * b^(4)) * c^(4)) + ((2 * b^(6)) * c^(4)) + ((2 * a^(4)) * c^(6)) + (((11 * a^(2)) * b^(2)) * c^(6)) + ((4 * b^(4)) * c^(6)) - ((3 * a^(2)) * c^(8)) - ((4 * b^(2)) * c^(8)) + c^(10))) * ((2 * a^(10)) - ((5 * a^(8)) * b^(2)) + ((2 * a^(6)) * b^(4)) + ((4 * a^(4)) * b^(6)) - ((4 * a^(2)) * b^(8)) + b^(10) - ((5 * a^(8)) * c^(2)) + (((10 * a^(6)) * b^(2)) * c^(2)) - (((7 * a^(4)) * b^(4)) * c^(2)) + (((11 * a^(2)) * b^(6)) * c^(2)) - ((3 * b^(8)) * c^(2)) + ((2 * a^(6)) * c^(4)) - (((7 * a^(4)) * b^(2)) * c^(4)) - (((14 * a^(2)) * b^(4)) * c^(4)) + ((2 * b^(6)) * c^(4)) + ((4 * a^(4)) * c^(6)) + (((11 * a^(2)) * b^(2)) * c^(6)) + ((2 * b^(4)) * c^(6)) - ((4 * a^(2)) * c^(8)) - ((3 * b^(2)) * c^(8)) + c^(10)))})
t = OrthogonalLine(x23096, i)
f_1 = OrthogonalLine(x23096, j)
g_1 = OrthogonalLine(x23096, k)
F = Intersect(k, g_1)
G = Intersect(j, f_1)
h_1 = Line(G, F)