10 Kreise im gegebenen Dreieck

Information / Quelle: Jürgen Köller, 13. 09. 2013

Arbeitsschritte

auch mit Schreibstift, Zirkel und Lineal darstellbar
- Dreieck mit s = 1.0 [LE] in GeoGebra konstruiert
- vorgegebene Kreise eingearbeitet
- Dreiecke bestimmt
- Radien der 4 Kreise berechnet

Gleichseitiges Dreieck mit Seite s = 1.0 [LE]:
Dreieckshöhe h = s/2 * √3 = 0.5 * √3 = 0.866025403784439 [LE] 

h = \overline{AJ}

R1

KP1

Legende: 
KP = Kontrollpunkte (KP1 bis KP5),
         zur Verifizierung ob Tangente durch Kontrollpunkt geht
         oder Kreis am Nachbarkreis anliegt, aber nicht schneidet .

KP2

KP3

KP4

KP5

Kreis 1

Dreieck 1

c1

a1

R1

Kreis 1:
             Dreieck 1
             R1 = c1 * sin 30° =
             = h - 0.5 * sin 30° = 0,366025403784439 * sin 30° = 
             R1 = 0,18301270189222 [LE]  

Dreieck 2

a2

b2

Dreieck 3

b3

a3

Kreis 2

R2

Kreis 2: 
             Dreieck 2
             a2 = 0.5 + R1 = 0,68301270189222
             b2 = a2 * tan 30° = 0,394337567297407              
             Dreieck 3
             b3 = 0.5 - b2 = 0,105662432702593
             a3 = b3 * tan 30° = 0,061004233964073 
             R2 = 0,061004233964073 [LE] 

Dreieck 4

a4

b4

c4

Kreis 3

Dreieck 5

a5

b5 = R1

c5 = c4/2

R3

Kreis 3:
             Dreieck 4
             c4 = √[R1² + (2R1)²] = 0,409228842176908
             Dreieck 5
             a5 = √[(c4/2)² - R1²] = 0,09150635094611 
             R3 = a5/2 = 0,045753175473055 [LE] 

Dreieck 6

b6 = R1

c6

Kreis 4

R4

Kreis 4:
             Dreieck 6
             c6 = R1/cos 30° = 0,211324865405188            
             R4 = c6 - R1 = 0,028312163512968 [LE]

P.K. September 2013

A = Intersect(xAxis, yAxis)
B = Point(xAxis)
C = Point(xAxis)
a = Segment(B, C)
D = Point(yAxis)
E = Point(yAxis)
b = Segment(D, E)
c = Circle(A, 0.5)
F = Intersect(c, a, 1)
G = Intersect(c, a, 2)
g_2 = CircleArc(A, G, F)
h_2 = Segment(F, G)
H = Intersect(c, b, 1)
d = Circle(F, Segment[A, F])
I = Intersect(c, d, 2)
e = Ray(F, I)
J = Intersect(b, e)
f = Segment(F, J)
g = Segment(J, G)
h = OrthogonalLine(H, f)
K = Intersect(h, f)
i = Segment(K, H)
α = Angle(J, K, H)
k = Circle(H, K)
L = Intersect(c, g, 2)
j = Ray(G, I)
M = Point(j)
l = Segment(G, M)
m = Ray(F, L)
N = Point(m)
n = Segment(F, N)
O = Intersect(k, l)
p = Line(H, g)
q = Line(H, f)
P = Intersect(p, l)
r = Line(P, a)
Q = Intersect(q, r)
s = Segment(H, Q)
t = Segment(Q, P)
a_1 = Segment(P, H)
c_1 = Circle(Q, Segment[H, K])
d_1 = Circle(P, Segment[H, K])
b_1 = Line(P, b)
R = Intersect(b_1, a)
g_1 = Segment(Q, R)
e_1 = Segment(P, R)
β = Angle(P, R, G)
S = Intersect(d_1, l, 2)
T = Intersect(b, l)
f_1 = Circle(T, Segment[T, S])
Z_1 = Intersect(f_1, c_1)
U = Intersect(d_1, g_1, 1)
h_1 = Ray(P, U)
V = Intersect(a, h_1)
i_1 = Segment(P, V)
W = Intersect(b, i_1)
k_1 = Circle(W, Segment[U, W])
Z = Intersect(c_1, k_1)
p_1 = Circle(L, Segment[W, Z])
A_1 = Intersect(p_1, n, 1)
q_1 = Circle(A_1, Segment[L, A_1])
r_1 = Circle(I, Segment[W, Z])
B_1 = Intersect(r_1, l, 1)
s_1 = Circle(B_1, Segment[I, B_1])
C_1 = Intersect(q_1, n, 1)
D_1 = Intersect(d_1, l, 1)
j_1 = Line(D_1, g)
E_1 = Intersect(j_1, f)
l_1 = Segment(D_1, E_1)
m_1 = OrthogonalLine(E_1, f)
F_1 = Intersect(m_1, b)
n_1 = Segment(E_1, F_1)
t_1 = Circle(F_1, Segment[F_1, E_1])
G_1 = Intersect(t_1, g)
a_2 = Line(F_1, g)
H_1 = Intersect(a_2, l)
b_2 = Segment(F_1, H_1)
c_2 = Line(H_1, a)
I_1 = Intersect(c_2, n)
d_2 = Segment(H_1, I_1)
e_2 = Circle(H_1, Segment[H_1, D_1])
f_2 = Circle(I_1, Segment[H_1, D_1])
J_1 = Intersect(e_2, g)
i_2 = OrthogonalLine(P, g)
γ = Angle(T, G, L)
k_2 = Segment(J, A)
Vieleck2, k_3, j_3, h_3 = Polygon(J, H, K)
δ = Angle(K, J, H)
l_2 = OrthogonalLine(H_1, a)
L_1 = Intersect(a, l_2)
m_2 = Segment(H_1, L_1)
M_1 = Intersect(k, t_1)
Vieleck1, l_3, a_3, m_3 = Polygon(A, M_1, L_1)
ε = Angle(A, M_1, L_1)
ζ = Angle(L_1, G, H_1)
Vieleck3, g_3, l_4, h_4 = Polygon(L_1, H_1, G)
N_1 = Intersect(k_1, g_1, 1)
Vieleck4, r_2, q_2, p_2 = Polygon(Q, P, R)
Vieleck5, n_3, r_3, a_4 = Polygon(R, A, N_1)
O_1 = Intersect(d_1, c_1)
Vieleck6, p_3, o_1, t_2 = Polygon(O_1, T, P)
P_1 = (0.4428359602853914, 0.011573543026582561)
Q_1 = (0.43259072370339324, -0.027202157269204844)
s_2 = Segment(P_1, Q_1)
b_3 = Segment(H_1, J_1)
R_1 = (-0.032478313925327705, 0.013777204922782574)
S_1 = (-0.045190699184579, -0.021893474792671817)
c_3 = Segment(R_1, S_1)
T_1 = (0.061916655101471874, 0.020264739376511347)
U_1 = (0.04903841477388234, -0.02056630417353856)
d_3 = Segment(T_1, U_1)
e_3 = Segment(W, Z)
η = Angle(T, P, O_1)
V_1 = Point(f_1)
W_1 = (-0.053039758874304885, 0.2942053065418411)
f_3 = Segment(V_1, W_1)
i_3 = Segment(T, Z_1)
A_2 = Point(n)
B_2 = (-0.039597619866403644, 0.2404367505102366)
q_3 = Segment(A_2, B_2)