Author: M Dabbs

This intersecting chords result underpins all of the complex number results that are included here.

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Result

Extra Lines

Therefore, triangles APC and DPB are similar (AAA)

Step 1

Step 2

Step 3

O = (6.3508441686031, 7.326299136819101)
R = (11.71233545158351, 1.9910545935984778)
c = Circle(O, R)
B = Point(c)
D = Point(c)
C = Point(c)
A = Point(c)
a = Segment(B, A)
b = Segment(D, C)
P = Intersect(a, b)
dAP = Distance(A, P)
dPB = Distance(P, B)
dCP = Distance(C, P)
dPD = Distance(P, D)
text1: "Product on Chord AB = AP * PB =" + (dAP * dPB)
text2: "Product on Chord CD = CP * PD =" + (dCP * dPD)
d = Segment(B, D)
e = Segment(C, A)
f = AngularBisector(A, P, D)
g = OrthogonalLine(D, f)
h = OrthogonalLine(B, f)
i = Line(D, C)
j = Line(B, A)
E = Intersect(h, g)
SFk: Distance[P, C] / Distance[B, P]
H = Intersect(h, f)
I = Intersect(f, g)
k = Line(B, C)
α = Angle(B, D, C)
β = Angle(B, A, C)
l = Line(D, A)
γ = Angle(A, B, D)
δ = Angle(A, C, D)
ε = Angle(D, P, B)
ζ = Angle(C, P, A)
m = true
p = false
dOPOR: Distance[O, R] - Distance[O, P]
n = false
o = false
q = false
text3: "\[AP = \frac {CP * PD} {PB}\ =" + (dCP * dPD / dPB) + "cm" + "\quad \[As can be seen from the direct measure in the Algebra View\]"

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')