Author: M Dabbs

Complex division using a construction

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Construction

\frac{A}{B}\

A_1 = Intersect(xAxis, yAxis)
B_1 = Point(yAxis)
A = (2.1858796414843087, 4.130350798369206)
B = (2.842229034665529, 0.5485583955802618)
b = Segment(A_1, B)
a = Segment(A_1, A)
c = Circle(A_1, A)
d = Circle(A_1, B)
C_1 = Intersect(d, xAxis, 2)
D = Intersect(c, xAxis, 2)
e = AngularBisector(A, A_1, D)
B' = Mirror(B, e)
f = Ray(A_1, B')
E = Intersect(c, yAxis, 1)
F = Intersect(d, xAxis, 1)
g = Circle(B_1, F, E)
C = Intersect(g, xAxis, 2)
h = Circle(A_1, C)
G = Intersect(h, f, 1)
i = Segment(A_1, G)
j = false

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')