Four Reflection for Half-Turn

Half-turn

R_4

R_3

R_2

R_1

R_{ab}

A

A'

A''

A = (-8.043669985718845, 2.493213541671999)
B = (-8.331266781678346, 5.632811897563261)
C = (-5.8866940160225685, 5.776610295543014)
D = (-6.270156410635241, 4.242760717092317)
E = (-7.2048459975036305, 4.554323912715114)
F = (-6.941215601207418, 2.6370119396517517)
poly1, a, b, c, d, e, f = Polygon(A, B, C, D, E, F)
G = (-9.602044944003506, -2.836551164646898)
H = Point(j)
h = Line(G, H)
I = (6.445970865187704, 5.082315226480167)
i = OrthogonalLine(I, h)
A' = Mirror(A, h)
B' = Mirror(B, h)
C' = Mirror(C, h)
D' = Mirror(D, h)
E' = Mirror(E, h)
F' = Mirror(F, h)
poly1', a', b', c', d', e', f' = Polygon(A', B', C', D', E', F')
A'' = Mirror(A', j)
B'' = Mirror(B', j)
C'' = Mirror(C', j)
D'' = Mirror(D', j)
E'' = Mirror(E', j)
F'' = Mirror(F', j)
poly1'', a'', b'', c'', d'', e'', f'' = Polygon(A'', B'', C'', D'', E'', F'')
A''_1 = Mirror(A', j)
B''_1 = Mirror(B', j)
a''_1 = Segment(A''_1, B''_1)
B''_2 = Mirror(B', j)
C''_1 = Mirror(C', j)
b''_1 = Segment(B''_2, C''_1)
C''_2 = Mirror(C', j)
D''_1 = Mirror(D', j)
c''_1 = Segment(C''_2, D''_1)
D''_2 = Mirror(D', j)
E''_1 = Mirror(E', j)
d''_1 = Segment(D''_2, E''_1)
E''_2 = Mirror(E', j)
F''_1 = Mirror(F', j)
e''_1 = Segment(E''_2, F''_1)
F''_2 = Mirror(F', j)
A''_2 = Mirror(A', j)
f''_1 = Segment(F''_2, A''_2)
A''' = Mirror(A'', g)
B''' = Mirror(B'', g)
C''' = Mirror(C'', g)
D''' = Mirror(D'', g)
E''' = Mirror(E'', g)
F''' = Mirror(F'', g)
poly1''', a''', b''', c''', d''', e''', f''' = Polygon(A''', B''', C''', D''', E''', F''')
A'''_1 = Mirror(A'', g)
B'''_1 = Mirror(B'', g)
a'''_1 = Segment(A'''_1, B'''_1)
B'''_2 = Mirror(B'', g)
C'''_1 = Mirror(C'', g)
b'''_1 = Segment(B'''_2, C'''_1)
C'''_2 = Mirror(C'', g)
D'''_1 = Mirror(D'', g)
c'''_1 = Segment(C'''_2, D'''_1)
D'''_2 = Mirror(D'', g)
E'''_1 = Mirror(E'', g)
d'''_1 = Segment(D'''_2, E'''_1)
E'''_2 = Mirror(E'', g)
F'''_1 = Mirror(F'', g)
e'''_1 = Segment(E'''_2, F'''_1)
F'''_2 = Mirror(F'', g)
A'''_2 = Mirror(A'', g)
f'''_1 = Segment(F'''_2, A'''_2)
J = Mirror(A''_1, g)
K = Mirror(B''_1, g)
k = Segment(J, K)
L = Mirror(A''_1, g)
M = Mirror(B''_1, g)
N = Mirror(B''_2, g)
O = Mirror(C''_1, g)
l = Segment(N, O)
P = Mirror(B''_2, g)
Q = Mirror(C''_1, g)
R = Mirror(C''_2, g)
S = Mirror(D''_1, g)
m = Segment(R, S)
T = Mirror(C''_2, g)
U = Mirror(D''_1, g)
V = Mirror(D''_2, g)
W = Mirror(E''_1, g)
n = Segment(V, W)
Z = Mirror(D''_2, g)
A_1 = Mirror(E''_1, g)
B_1 = Mirror(E''_2, g)
C_1 = Mirror(F''_1, g)
p = Segment(B_1, C_1)
D_1 = Mirror(E''_2, g)
E_1 = Mirror(F''_1, g)
F_1 = Mirror(F''_2, g)
G_1 = Mirror(A''_2, g)
q = Segment(F_1, G_1)
H_1 = Mirror(F''_2, g)
I_1 = Mirror(A''_2, g)
J_1 = Mirror(A''', i)
K_1 = Mirror(B''', i)
L_1 = Mirror(C''', i)
M_1 = Mirror(D''', i)
N_1 = Mirror(E''', i)
O_1 = Mirror(F''', i)
poly2, j_1, k_1, l_1, m_1, n_1, o_1 = Polygon(J_1, K_1, L_1, M_1, N_1, O_1)
P_1 = Mirror(A''', i)
Q_1 = Mirror(B''', i)
r = Segment(P_1, Q_1)
R_1 = Mirror(B''', i)
S_1 = Mirror(C''', i)
s = Segment(R_1, S_1)
T_1 = Mirror(C''', i)
U_1 = Mirror(D''', i)
t = Segment(T_1, U_1)
V_1 = Mirror(D''', i)
W_1 = Mirror(E''', i)
a_1 = Segment(V_1, W_1)
Z_1 = Mirror(E''', i)
A_2 = Mirror(F''', i)
b_1 = Segment(Z_1, A_2)
B_2 = Mirror(F''', i)
C_2 = Mirror(A''', i)
c_1 = Segment(B_2, C_2)
D_2 = Mirror(A'''_1, i)
E_2 = Mirror(B'''_1, i)
d_1 = Segment(D_2, E_2)
F_2 = Mirror(A'''_1, i)
G_2 = Mirror(B'''_1, i)
H_2 = Mirror(B'''_2, i)
I_2 = Mirror(C'''_1, i)
e_1 = Segment(H_2, I_2)
J_2 = Mirror(B'''_2, i)
K_2 = Mirror(C'''_1, i)
L_2 = Mirror(C'''_2, i)
M_2 = Mirror(D'''_1, i)
f_1 = Segment(L_2, M_2)
N_2 = Mirror(C'''_2, i)
O_2 = Mirror(D'''_1, i)
P_2 = Mirror(D'''_2, i)
Q_2 = Mirror(E'''_1, i)
g_1 = Segment(P_2, Q_2)
R_2 = Mirror(D'''_2, i)
S_2 = Mirror(E'''_1, i)
T_2 = Mirror(E'''_2, i)
U_2 = Mirror(F'''_1, i)
h_1 = Segment(T_2, U_2)
V_2 = Mirror(E'''_2, i)
W_2 = Mirror(F'''_1, i)
Z_2 = Mirror(F'''_2, i)
A_3 = Mirror(A'''_2, i)
i_1 = Segment(Z_2, A_3)
B_3 = Mirror(F'''_2, i)
C_3 = Mirror(A'''_2, i)
J' = Mirror(J, i)
K' = Mirror(K, i)
k' = Segment(J', K')
J'_1 = Mirror(J, i)
K'_1 = Mirror(K, i)
L' = Mirror(L, i)
M' = Mirror(M, i)
N' = Mirror(N, i)
O' = Mirror(O, i)
l' = Segment(N', O')
N'_1 = Mirror(N, i)
O'_1 = Mirror(O, i)
P' = Mirror(P, i)
Q' = Mirror(Q, i)
R' = Mirror(R, i)
S' = Mirror(S, i)
m' = Segment(R', S')
R'_1 = Mirror(R, i)
S'_1 = Mirror(S, i)
T' = Mirror(T, i)
U' = Mirror(U, i)
V' = Mirror(V, i)
W' = Mirror(W, i)
n' = Segment(V', W')
V'_1 = Mirror(V, i)
W'_1 = Mirror(W, i)
Z' = Mirror(Z, i)
D_3 = Mirror(A_1, i)
E_3 = Mirror(B_1, i)
F_3 = Mirror(C_1, i)
p' = Segment(E_3, F_3)
G_3 = Mirror(B_1, i)
H_3 = Mirror(C_1, i)
I_3 = Mirror(D_1, i)
J_3 = Mirror(E_1, i)
K_3 = Mirror(F_1, i)
L_3 = Mirror(G_1, i)
q' = Segment(K_3, L_3)
M_3 = Mirror(F_1, i)
N_3 = Mirror(G_1, i)
O_3 = Mirror(H_1, i)
P_3 = Mirror(I_1, i)
Q_3 = Intersect(i, g)
p_1 = Circle(H, 1)
q_1 = Circle(Q_3, 1)
R_3 = Intersect(p_1, j, 2)
S_3 = Intersect(q_1, g, 2)
T_3 = Intersect(p_1, h, 1)
U_3 = Intersect(q_1, i, 1)
α = Angle(T_3, H, R_3)
β = Angle(S_3, Q_3, U_3)
r_1 = Line(H, Q_3)
V_3 = Intersect(p_1, r_1, 2)
W_3 = Intersect(q_1, r_1, 1)
Z_3 = Intersect(h, i)
γ = Angle(H, Z_3, Q_3)
A_4 = Rotate(V_3, -α, H)
δ = Angle(A_4, H, V_3)
B_4 = Rotate(W_3, β, Q_3)
ε = Angle(W_3, Q_3, B_4)
s_1 = Line(H, A_4)
t_1 = Line(Q_3, B_4)
C_4 = Intersect(s_1, t_1)
c_2 = Circle(Q_3, Z_3, H)
d_2 = Circle(C_4, 0.5)
D_4 = Intersect(d_2, s_1, 2)
E_4 = Intersect(d_2, t_1, 2)
e_2 = Circle(H, 1)
f_2 = Circle(Q_3, 1)
F_4 = Intersect(e_2, r_1, 1)
G_4 = Intersect(f_2, r_1, 2)
h_2 = Segment(B, C)
i_2 = Segment(C, D)
j_2 = Segment(D, E)
k_2 = Segment(E, F)
l_2 = Segment(F, A)
m_2 = Segment(A, B)
p_2 = Segment(B, C)
q_2 = Segment(C, D)
r_2 = Segment(D, E)
s_2 = Segment(E, F)
t_2 = Segment(F, A)
a_3 = Segment(A, B)
b_3 = Segment(O_3, P_3)
c_3 = Segment(P_3, P')
d_3 = Segment(P', T')
e_3 = Segment(T', Z')
f_3 = Segment(Z', I_3)
g_3 = Segment(I_3, O_3)
o = false
h_3 = Semicircle(T', C)
H_4 = (-3.921449243632624, 9.94676383695582)
p_3 = Circle(H_4, 1)
n_2 = OrthogonalLine(H_4, j)
I_4 = Intersect(p_3, n_2, 2)
J_4 = Intersect(p_3, n_2, 1)
q_3 = Semicircle(J_4, I_4)
K_4 = Point(q_3)
i_3 = Segment(H_4, K_4)
j_3 = Segment(I_4, H_4)
r_3 = CircleArc(H_4, I_4, K_4)
s_3 = CircleSector(H_4, I_4, K_4)
ζ = Angle(I_4, H_4, K_4)
A'_1 = Rotate(A, ζ, C_4)
B'_1 = Rotate(B, ζ, C_4)
C'_1 = Rotate(C, ζ, C_4)
D'_1 = Rotate(D, ζ, C_4)
E'_1 = Rotate(E, ζ, C_4)
F'_1 = Rotate(F, ζ, C_4)
poly1'_1, a'_1, b'_1, c'_1, d'_1, e'_1, f'_1 = Polygon(A'_1, B'_1, C'_1, D'_1, E'_1, F'_1)
B'_2 = Rotate(B, ζ, C_4)
C'_2 = Rotate(C, ζ, C_4)
l_3 = Segment(B'_2, C'_2)
C'_3 = Rotate(C, ζ, C_4)
D'_2 = Rotate(D, ζ, C_4)
m_3 = Segment(C'_3, D'_2)
D'_3 = Rotate(D, ζ, C_4)
E'_2 = Rotate(E, ζ, C_4)
n_3 = Segment(D'_3, E'_2)
E'_3 = Rotate(E, ζ, C_4)
F'_2 = Rotate(F, ζ, C_4)
t_3 = Segment(E'_3, F'_2)
F'_3 = Rotate(F, ζ, C_4)
A'_2 = Rotate(A, ζ, C_4)
a_4 = Segment(F'_3, A'_2)
A'_3 = Rotate(A, ζ, C_4)
B'_3 = Rotate(B, ζ, C_4)
b_4 = Segment(A'_3, B'_3)
B'_4 = Rotate(B, ζ, C_4)
C'_4 = Rotate(C, ζ, C_4)
c_4 = Segment(B'_4, C'_4)
C'_5 = Rotate(C, ζ, C_4)
D'_4 = Rotate(D, ζ, C_4)
d_4 = Segment(C'_5, D'_4)
D'_5 = Rotate(D, ζ, C_4)
E'_4 = Rotate(E, ζ, C_4)
e_4 = Segment(D'_5, E'_4)
E'_5 = Rotate(E, ζ, C_4)
F'_4 = Rotate(F, ζ, C_4)
f_4 = Segment(E'_5, F'_4)
F'_5 = Rotate(F, ζ, C_4)
A'_4 = Rotate(A, ζ, C_4)
g_4 = Segment(F'_5, A'_4)
A'_5 = Rotate(A, ζ, C_4)
B'_5 = Rotate(B, ζ, C_4)
h_4 = Segment(A'_5, B'_5)
j_4 = Segment(C'_5, D'_5)
k_4 = Segment(D'_5, E'_5)
l_4 = Segment(E'_5, F'_5)
m_4 = Segment(F'_5, A'_5)
n_4 = Segment(A'_5, B'_5)
p_4 = Segment(B'_5, C'_5)
i_4 = Segment(C_4, C'_5)
u = false
v = false
w = false
z = false
k_3 = Line(C_4, i)
q_4 = OrthogonalLine(C_4, k_3)
L_4 = Intersect(k_3, q_4)
A'_6 = Mirror(A, q_4)
B'_6 = Mirror(B, q_4)
C'_6 = Mirror(C, q_4)
D'_6 = Mirror(D, q_4)
E'_6 = Mirror(E, q_4)
F'_6 = Mirror(F, q_4)
poly1'_2, a'_6, b'_6, c'_6, d'_6, e'_6, f'_6 = Polygon(A'_6, B'_6, C'_6, D'_6, E'_6, F'_6)
B'_7 = Mirror(B, q_4)
C'_7 = Mirror(C, q_4)
r_4 = Segment(B'_7, C'_7)
C'_8 = Mirror(C, q_4)
D'_7 = Mirror(D, q_4)
s_4 = Segment(C'_8, D'_7)
D'_8 = Mirror(D, q_4)
E'_7 = Mirror(E, q_4)
t_4 = Segment(D'_8, E'_7)
E'_8 = Mirror(E, q_4)
F'_7 = Mirror(F, q_4)
a_5 = Segment(E'_8, F'_7)
F'_8 = Mirror(F, q_4)
A'_7 = Mirror(A, q_4)
b_5 = Segment(F'_8, A'_7)
A'_8 = Mirror(A, q_4)
B'_8 = Mirror(B, q_4)
c_5 = Segment(A'_8, B'_8)
B'_9 = Mirror(B, q_4)
C'_9 = Mirror(C, q_4)
d_5 = Segment(B'_9, C'_9)
C'_{10} = Mirror(C, q_4)
D'_9 = Mirror(D, q_4)
e_5 = Segment(C'_{10}, D'_9)
D'_{10} = Mirror(D, q_4)
E'_9 = Mirror(E, q_4)
f_5 = Segment(D'_{10}, E'_9)
E'_{10} = Mirror(E, q_4)
F'_9 = Mirror(F, q_4)
g_5 = Segment(E'_{10}, F'_9)
F'_{10} = Mirror(F, q_4)
A'_9 = Mirror(A, q_4)
h_5 = Segment(F'_{10}, A'_9)
A'_{10} = Mirror(A, q_4)
B'_{10} = Mirror(B, q_4)
i_5 = Segment(A'_{10}, B'_{10})
M_4 = Mirror(A'_6, k_3)
N_4 = Mirror(B'_6, k_3)
O_4 = Mirror(C'_6, k_3)
P_4 = Mirror(D'_6, k_3)
Q_4 = Mirror(E'_6, k_3)
R_4 = Mirror(F'_6, k_3)
poly3, m_5, n_5, o_4, p_5, q_5, r_5 = Polygon(M_4, N_4, O_4, P_4, Q_4, R_4)
S_4 = Mirror(A'_6, k_3)
T_4 = Mirror(B'_6, k_3)
j_5 = Segment(S_4, T_4)
U_4 = Mirror(B'_6, k_3)
V_4 = Mirror(C'_6, k_3)
k_5 = Segment(U_4, V_4)
W_4 = Mirror(C'_6, k_3)
Z_4 = Mirror(D'_6, k_3)
l_5 = Segment(W_4, Z_4)
A_5 = Mirror(D'_6, k_3)
B_5 = Mirror(E'_6, k_3)
s_5 = Segment(A_5, B_5)
C_5 = Mirror(E'_6, k_3)
D_5 = Mirror(F'_6, k_3)
t_5 = Segment(C_5, D_5)
E_5 = Mirror(F'_6, k_3)
F_5 = Mirror(A'_6, k_3)
a_6 = Segment(E_5, F_5)
G_5 = Mirror(B'_7, k_3)
H_5 = Mirror(C'_7, k_3)
b_6 = Segment(G_5, H_5)
I_5 = Mirror(B'_7, k_3)
J_5 = Mirror(C'_7, k_3)
K_5 = Mirror(C'_8, k_3)
L_5 = Mirror(D'_7, k_3)
c_6 = Segment(K_5, L_5)
M_5 = Mirror(C'_8, k_3)
N_5 = Mirror(D'_7, k_3)
O_5 = Mirror(D'_8, k_3)
P_5 = Mirror(E'_7, k_3)
d_6 = Segment(O_5, P_5)
Q_5 = Mirror(D'_8, k_3)
R_5 = Mirror(E'_7, k_3)
S_5 = Mirror(E'_8, k_3)
T_5 = Mirror(F'_7, k_3)
e_6 = Segment(S_5, T_5)
U_5 = Mirror(E'_8, k_3)
V_5 = Mirror(F'_7, k_3)
W_5 = Mirror(F'_8, k_3)
Z_5 = Mirror(A'_7, k_3)
f_6 = Segment(W_5, Z_5)
A_6 = Mirror(F'_8, k_3)
B_6 = Mirror(A'_7, k_3)
C_6 = Mirror(A'_8, k_3)
D_6 = Mirror(B'_8, k_3)
g_6 = Segment(C_6, D_6)
E_6 = Mirror(A'_8, k_3)
F_6 = Mirror(B'_8, k_3)
G_6 = Mirror(B'_9, k_3)
H_6 = Mirror(C'_9, k_3)
h_6 = Segment(G_6, H_6)
I_6 = Mirror(B'_9, k_3)
J_6 = Mirror(C'_9, k_3)
K_6 = Mirror(C'_{10}, k_3)
L_6 = Mirror(D'_9, k_3)
i_6 = Segment(K_6, L_6)
M_6 = Mirror(C'_{10}, k_3)
N_6 = Mirror(D'_9, k_3)
O_6 = Mirror(D'_{10}, k_3)
P_6 = Mirror(E'_9, k_3)
j_6 = Segment(O_6, P_6)
Q_6 = Mirror(D'_{10}, k_3)
R_6 = Mirror(E'_9, k_3)
S_6 = Mirror(E'_{10}, k_3)
T_6 = Mirror(F'_9, k_3)
k_6 = Segment(S_6, T_6)
U_6 = Mirror(E'_{10}, k_3)
V_6 = Mirror(F'_9, k_3)
W_6 = Mirror(F'_{10}, k_3)
Z_6 = Mirror(A'_9, k_3)
l_6 = Segment(W_6, Z_6)
A_7 = Mirror(F'_{10}, k_3)
B_7 = Mirror(A'_9, k_3)
C_7 = Mirror(A'_{10}, k_3)
D_7 = Mirror(B'_{10}, k_3)
m_6 = Segment(C_7, D_7)
E_7 = Mirror(A'_{10}, k_3)
F_7 = Mirror(B'_{10}, k_3)
n_6 = Segment(C, C'_{10})
p_6 = Segment(C'_{10}, M_6)
q_6 = Segment(A, A'_{10})
r_6 = Segment(B'_{10}, F_7)
u_1 = false
t_6 = Segment(A_7, E_7)
a_7 = Segment(E_7, F_7)
b_7 = Segment(F_7, M_6)
c_7 = Segment(M_6, Q_6)
d_7 = Segment(Q_6, U_6)
e_7 = Segment(U_6, A_7)
f_7 = Segment(D'_{10}, C'_{10})
g_7 = Segment(C'_{10}, B'_{10})
h_7 = Segment(B'_{10}, A'_{10})
i_7 = Segment(A'_{10}, F'_{10})
j_7 = Segment(F'_{10}, E'_{10})
k_7 = Segment(E'_{10}, D'_{10})
η = Angle(H, Q_3, C_4)
G_7 = Point(g)
θ = Angle(G_7, Q_3, I)
ι = Angle(C_4, H, Q_3)
H_7 = Intersect(h_3, j, 1)
κ = Angle(G, H, H_7)
b_2 = Segment(C_4, Q_3)
a_2 = Segment(C_4, H)
g_2 = Segment(H, Q_3)

Tool: Complex Multiplication (command ComplexMultiplication)
Select Points at O(0, 0), P(0, -1), A and B for A*B
Input: O, P, A, B
Output: j, AB

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
b = Segment(O, A)
f = Segment(O, B)
G = Intersect(d, xAxis, 1)
g = Circle(D, P, G)
E = Intersect(g, yAxis, 2)
e = Circle(O, E)
a, h = AngularBisector(f, b)
D' = Mirror(D, a)
i = Ray(O, D')
AB = Intersect(e, i, 1)
j = Segment(O, AB)

Tool: Complex Division (command ComplexDivision)
Select Points O(0, 0), P(0, -1), A and B for A/B
Input: O, P, A, B
Output: l, C

A/B

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
e = AngularBisector(A, O, D)
B' = Mirror(B, e)
f = Ray(O, B')
E = Intersect(c, yAxis, 1)
F = Intersect(d, xAxis, 1)
g = Segment(F, E)
G = Midpoint(g)
h = OrthogonalLine(G, g)
i = Segment(P, F)
H = Midpoint(i)
j = OrthogonalLine(H, i)
I = Intersect(h, j)
k = Circle(I, F)
J = Intersect(k, xAxis, 2)
p = Circle(O, J)
C = Intersect(p, f)
l = Segment(O, C)

Tool: Complex Addition (command ComplexAddition)
Select Points O(0, 0), A and B for A + B
Input: O, A, B
Output: C', c

A + B

u = Vector(O, B)
C' = Translate(A, u)
c = Segment(O, C')

Tool: Complex Subtraction (command ComplexSubtraction)
Select Points O(0, 0), A and B for A - B
Input: O, A, B
Output: c, A'

A - B

u = Vector(B, O)
A'_1 = Translate(A, u)
A' = Translate(A, u)
c = Segment(O, A')