equazione\ ax =\sqrt{x}

rettangolo equivalente AB x BT

 AE=\frac{19}{20}AB- (XII.3.7,8)

 AE=\frac{1}{20}AB-(XII.3.9)

AE=\frac{39}{20}AB- (XII.3.10)

 AE=\frac{21}{20}AB-(XII.3.11)

 AE=\frac{41}{20}AB-(XII.3.12)

i \ “casi”\  di\  Fibonacci

XII.3.7,8

XII.3.9

XII.3.10

XII.3.11

XII.3.12

Poniamo\ AB=5\ AE=aAB , \AE×AE=AB×BT \ \ \ ⇒ \ \ a\frac{AB}{BT}=\sqrt{\frac{AB}{BT}}

AE x AE

1⇒

2⇒

A: (-1, 0)
B: (4, 0)
AB = Segment(A, B)
a: {19, 1, 39, 21, 41}
n: 5
txa = FractionText(Element[a, n] / 20)
txta: "a=" + (LaTeX[txa]) + ""
k: Element[a, n] / 20
E = Point(Circle[A, (k * AB)])
AE = Segment(A, E)
E' = Mirror(E, A)
poli1, f, g, h, i, C, D = Polygon(E', A, 4)
txtAE2 = FractionText(poli1)
txAE2: "AE^{2}=" + (LaTeX[txtAE2]) + ""
testo2 = 1.7 (slider from -5 to 5)
j = OrthogonalLine(B, AE)
F = Intersect(h, j)
l = Line(F, A)
G = Intersect(i, l)
m = OrthogonalLine(G, j)
T = Intersect(j, m)
B' = Mirror(B, Midpoint[A, T])
q1, a_1, b', BT, b = Polygon(A, B', T, B)
txtRet = FractionText(q1)
txABxBT: "AB×BT=" + (LaTeX[txtRet]) + ""
s = 1 (slider from 0 to 1, step 0.01)
u = Vector(C, T)
H = Point(BT)
v = Vector(D, H)
J = Point(AE)
w = Vector(E', A)
c = Vector(A, B')
A' = Translate(A, Vector[(s * c)])
E'_{1} = Translate(E', Vector[(s * w)])
D' = Translate(D, Vector[(s * v)])
C' = Translate(C, Vector[(s * u)])
q2, a', c', d', e'_{1} = Polygon(A', C', D', E'_{1})
d = true
K = (4.93893129770992, 5.362595419847329)
xx: AB / BT
txtXX = FractionText(xx)
txtAE = FractionText(AE)
txtAB = FractionText(AB)
txtBT = FractionText(BT)
txX: "x=\frac{AB}{BT}=\frac{" + (LaTeX[txtAB]) + "}{" + (LaTeX[txtBT]) + "}=" + (LaTeX[txtXX]) + ""
txt1sua2 = FractionText((1 / k)^(2))
txt1sua = FractionText(1 / k)
testo3: "x={\left(  \frac{1}{a}  \right)}^2=\left( " + (LaTeX[txt1sua]) + " \right) ^2=" + (LaTeX[txt1sua2]) + ""
rot = 1 (slider from 0 to 1, step 0.01)
A'_{1} = Rotate(A, 90°, A)
E'_{2} = Rotate(E, 90°, A)
AE' = Segment(A'_{1}, E'_{2})
e = Vector(A, E')
L = Translate(A'_{1}, Vector[(rot * e)])
M = Translate(E'_{2}, Vector[(rot * e)])
AE'' = Segment(L, M)
A'_{2} = Rotate(A, (rot * 90°), A)
E'_{3} = Rotate(E, (rot * 90°), A)
AE'_{1} = Segment(A'_{2}, E'_{3})
testo5: "BT=" + (LaTeX[txtBT]) + ""