S = S_1 = S_2

A = (5.510881270649511, 6.166335541507631)
B = (1.9153018410296134, 1.3936318938574097)
C = (8.250370359883718, 1.3294251183284826)
t1, c, a, b = Polygon(A, B, C)
f = AngularBisector(C, A, B)
g = AngularBisector(A, B, C)
d = Incircle(A, B, C)
D = Intersect(d, a, 1)
E = Intersect(d, b, 1)
F = Intersect(Circle[C, Segment[B, D]], a)
G = Intersect(Circle[C, Segment[A, E]], b)
h = OrthogonalLine(F, a)
i = OrthogonalLine(G, b)
H_1 = Intersect(f, h)
I_b = Intersect(g, i)
e = Circle(H_1, Segment[F, H_1])
p = Circle(I_b, Segment[I_b, G])
j = Line(A, B)
P = Intersect(e, j, 2)
k = Line(A, C)
Q = Intersect(e, k, 2)
l = Line(B, C)
M = Intersect(p, j, 1)
N = Intersect(p, l, 2)
m = Segment(M, N)
n = OrthogonalLine(C, Segment[M, N])
K = Intersect(n, m)
q = Segment(P, Q)
L = Intersect(q, OrthogonalLine[C, q])
r = Segment(C, L)
s = Segment(C, K)
α = Angle(C, K, N)
β = Angle(C, L, P)
I = Intersect(d, j, 1)
J = Intersect(Circle[B, Segment[A, I]], c)
t = OrthogonalLine(J, c)
f_1 = AngularBisector(A, C, B)
I_c = Intersect(t, f_1)
g_1 = OrthogonalLine(I_c, a)
T = Intersect(l, g_1)
c_1 = Circle(I_c, T)
d_1 = Circle(I_b, N)
S = Intersect(Line[M, N], Line[P, Q])
h_1 = Segment(P, S)
i_1 = Segment(N, S)
X = Intersect(OrthogonalLine[M, Segment[P, Q]], Segment[P, Q])
Y = Intersect(OrthogonalLine[P, Segment[M, N]], Segment[M, N])
j_1 = Segment(M, X)
k_1 = Segment(P, Y)
H = Intersect(j_1, k_1)
γ = Angle(P, Y, S)
δ = Angle(M, X, P)
l_1 = Segment(H, S)