Two Rotations Different Centres

α/2

β/2

γ = α + β

$ \frac{\alpha}{2} $

$ \frac{β}{2} $

Construction

2nd Rotation

1st Rotation

Property

A = (-4.416685251120368, -5.658829071579154)
B = (-4.248920453477323, -2.111801921411938)
C = (-1.972112485464581, -2.111801921411938)
D = (-1.7803812881582446, -3.549785901209462)
E = (-3.290264466945634, -3.7894498978423825)
F = (-3.1224996693025884, -5.754694670232324)
poly1, a, b, c, d, e, f = Polygon(A, B, C, D, E, F)
G = (-4.202150245979608, 6.075152586870539)
g = Circle(G, 1)
H = (-4.111312911985794, 3.5408699010499314)
h = Circle(H, 1)
i = OrthogonalLine(G, yAxis)
j = OrthogonalLine(H, yAxis)
I = Intersect(g, i, 1)
J = Intersect(h, j, 1)
K = Point(g)
L = Point(h)
k = Segment(G, I)
l = Segment(G, K)
m = Segment(H, L)
n = Segment(H, J)
α = Angle(I, G, K)
β = Angle(J, H, L)
M = (0.0672044517296486, 1.6846814438361946)
A' = Rotate(A, α, M)
B' = Rotate(B, α, M)
C' = Rotate(C, α, M)
D' = Rotate(D, α, M)
E' = Rotate(E, α, M)
F' = Rotate(F, α, M)
poly1', a', b', c', d', e', f' = Polygon(A', B', C', D', E', F')
N = (8.272843622504187, 5.6815241395640115)
A'' = Rotate(A', β, N)
B'' = Rotate(B', β, N)
C'' = Rotate(C', β, N)
D'' = Rotate(D', β, N)
E'' = Rotate(E', β, N)
F'' = Rotate(F', β, N)
poly1'', a'', b'', c'', d'', e'', f'' = Polygon(A'', B'', C'', D'', E'', F'')
p = Line(M, N)
p' = Rotate(p, α / 2, M)
p'' = Mirror(p', p)
p'_1 = Rotate(p, β / 2, N)
O = Intersect(p'_1, p'')
γ_1 = Angle(p'', p)
δ = Angle(M, N, O)
P = (-4.050754689323251, 1.0368663265605955)
q = Circle(P, 1)
r = OrthogonalLine(P, yAxis)
R = Intersect(q, r, 1)
s = Segment(R, P)
R' = Rotate(R, α + β, P)
t_1 = CircleArc(P, R, R')
Q = Point(t_1)
a_1 = Segment(P, Q)
γ = Angle(R, P, Q)
A'_1 = Rotate(A, γ, O)
B'_1 = Rotate(B, γ, O)
C'_1 = Rotate(C, γ, O)
D'_1 = Rotate(D, γ, O)
E'_1 = Rotate(E, γ, O)
F'_1 = Rotate(F, γ, O)
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)
dist = Distance(C'_1, C'')
o = false
u = false
b_1 = Segment(M, B)
c_1 = Segment(M, B')
d_1 = Segment(F', N)
e_1 = Segment(N, F'')
g_1 = Segment(O, C)
h_1 = Segment(O, C'_1)
v = false
t = false
ε = Angle(C, O, C'_1)
ζ = Angle(B, M, B')
η = Angle(F', N, F'')
DistRot = Distance(B'_1, B'')

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')