9 Kreise im gegebenen Quadrat

Information / Quelle: Jürgen Köller, 21. 09. 2013

Arbeitsschritte

auch mit Schreibstift, Zirkel und Lineal darstellbar
- Quadrat mit s = 1.0 [LE] in GeoGebra konstruiert
- vorgegebene Kreise eingearbeitet
- Dreiecke bestimmt
- Radien der 3 Kreise berechnet

Quadratseite s = 1.0 [LE]

Wh

Legende: 
Wh = Winkelhalbierende
KP = Kontrollpunkt zur Verifizierung ob Kreis am
          Nachbarkreis anliegt, aber ihn nicht schneidet .

Kreis 1

Kreis 2

Kreis 3

KP1

KP

Dreieck 1

a1

R1

c1

Kreis 1
Dreieck 1
R1 = a1 * tanα = 0,207106781186548 [LE]

Dreieck 2

R2

b2

c2

Kreis 2
Dreieck 2
R2 = 0,5 - 2R1 = 0,085786437626905 [LE]
b2 = R1 + R2 = 0,292893218813452
c2 = √(b2² + R2²) = 0,305197887456024
β = atan(R2 / b2) = 16,32494993689524°

Dreieck 3

= c3

R1

a3

Kreis 3

Dreieck 3
a3 = √(c3² - R1²) = 0,224170764583983
c3 = c2
ε = asin(a3 / c3) = 47,26579039910019°
γ = ε - β = 30,94084046220495°

Dreieck 4

a4

b4

R3

Dreieck 4
a4 = 0,5 + R2 = 0,585786437626905 
δ = γ
b4 = a4 * tan δ = 0,351153302357084
R3 = 0,5 - b4 = 0.148846697642916 [LE]  

P.K. September 2013

A = Intersect(xAxis, yAxis)
B = Point(xAxis)
a = Segment(A, B)
C = Point(yAxis)
b = Segment(C, A)
c = Circle(A, B)
D = Intersect(c, b, 1)
d = AngularBisector(D, A, B)
e = Line(D, a)
f = Line(B, b)
E = Intersect(d, e)
g = Segment(D, E)
h = Segment(E, B)
i = Segment(D, A)
j = Segment(D, B)
k = Segment(A, E)
F = Intersect(j, k)
l = AngularBisector(D, E, F)
m = AngularBisector(E, D, F)
H = Intersect(l, j)
n = Segment(E, H)
J = Midpoint(D, E)
K = Midpoint(A, B)
L = Midpoint(E, B)
M = Midpoint(A, D)
q = Segment(J, K)
G = Intersect(l, m)
r = Segment(M, L)
s = Circle(G, J)
a_1 = OrthogonalLine(G, j)
c_1 = Circle(F, G)
N = Intersect(c_1, r, 2)
O = Intersect(c_1, q, 2)
P = Intersect(c_1, r, 1)
d_1 = Circle(P, M)
e_1 = Circle(N, L)
f_1 = Circle(O, K)
Q = Intersect(s, q, 2)
g_1 = Circle(F, Q)
R = Intersect(f_1, g_1, 1)
b_1 = Segment(R, P)
S = Midpoint(R, P)
h_1 = CircleArc(S, R, P)
U = Intersect(d_1, h_1, 1)
j_1 = Ray(R, U)
W = Intersect(g, j_1)
k_1 = Segment(W, R)
l_1 = Line(W, i)
A_1 = Intersect(l_1, j)
B_1 = Intersect(l_1, k)
m_1 = Segment(W, B_1)
p_1 = Circle(A_1, W)
C_1 = Intersect(p_1, d_1)
n_1 = Line(A_1, g)
q_1 = Line(B_1, g)
T = Intersect(n_1, k)
D_1 = Intersect(q_1, j)
r_1 = Segment(A_1, T)
s_1 = Segment(T, D_1)
t_1 = Segment(D_1, B_1)
c_2 = Circle(T, Segment[A_1, W])
d_2 = Circle(D_1, Segment[A_1, W])
e_2 = Circle(B_1, Segment[A_1, W])
Vieleck1, e_3, g_2, j_2 = Polygon(G, J, E)
t = OrthogonalLine(G, k)
α = Angle(J, E, G)
Vieleck2, f_2, p_2, r_2 = Polygon(P, R, F)
β = Angle(F, P, R)
Vieleck3, r_3, p_3, u_1 = Polygon(P, U, R)
ζ = Angle(P, U, R)
ε = Angle(U, P, R)
γ = Angle(U, P, F)
Vieleck4, j_3, r_4, w = Polygon(R, W, J)
G_1 = Point(g_1)
δ = Angle(J, R, W)
E_1 = Intersect(n_1, i)
a_2 = Segment(A_1, E_1)