Author: Andreas Brinken

   Ceva-

Gleichung

Teilverhält-

   nisse

Ecktrans-

I  =  Inkreismittelpunkt (X_1)

U = Umkreismittelpunkt (X_3)

H = Höhenschnittpunkt (X_4)

M = Mittelpunkt Feuerbachkreis (X_5)

N = Nagelpunkt (X_8)

Mi = Mittenpunkt (X_9)

Sp = Spiekerpunkt (X_{10})

F_1, F_2 = 1. und 2. Fermatpunkt (X_{13}, X_{14})

G = Gergoinnepunkt (X_7)

X, Y, Z = Ankreismittelpunkte

S = Schwerpunkt (X_2)

Schnittpunkte Seiten-Winkelhalbierende

      versale
(AA'), (BB'),(CC')

L = Lemoinepunkt (X_6)

Höhenfußpunkte

Verändere die Lage der Punkte A', B' und C' so, dass sich die drei 
Ecktransversalen in einem Punkt schneiden.

Oberer Höhenabschnitte (Eulerpunkte)

B = (0.9107897915067535, 1.3634334171420865)
C = (0.6385675975815205, 3.867877601254231)
e = Line(B, C)
A' = Point(e)
Verhältnis_a: Distance[B, A'] / Distance[A', C]
A = (-2.3195802430726786, 1.4178778559271332)
d = Line(A, B)
C' = Point(d)
Verhältnis_c: Distance[A, C'] / Distance[C', B]
f = Line(C, A)
B' = Point(f)
Verhältnis_b: Distance[C, B'] / Distance[B', A]
Produkt_{neg}: Verhältnis_a Verhältnis_b Verhältnis_c (-1)
AA1 = (-4.7695799883997765, 6.844173588170112)
Abstand_{ }Schieberegler = 2.5 (slider from 0.5 to 3, step 0.1)
NM11: (x(AA1) + 0.5, y(AA1))
NM22: (x(AA1) + Abstand_{ }Schieberegler + 0.5, y(AA1))
NM33: (x(AA1) + 2 Abstand_{ }Schieberegler + 0.5, y(AA1))
NM44: (x(AA1) + 3 Abstand_{ }Schieberegler + 0.5, y(AA1))
AA2: (x(AA1) + Abstand_{ }Schieberegler, y(AA1))
AA3: (x(AA1) + 2 Abstand_{ }Schieberegler, y(AA1))
AA4: (x(AA1) + 3 Abstand_{ }Schieberegler, y(AA1))
EE1: (x(AA1) + 1, y(AA1))
EE2: (x(AA1) + Abstand_{ }Schieberegler + 1, y(AA1))
EE3: (x(AA1) + 2 Abstand_{ }Schieberegler + 1, y(AA1))
EE4: (x(AA1) + 3 Abstand_{ }Schieberegler + 1, y(AA1))
mm1 = Segment(AA1, EE1)
mm2 = Segment(AA2, EE2)
mm3 = Segment(AA3, EE3)
mm4 = Segment(AA4, EE4)
MM1 = Point(mm1)
MM2 = Point(mm2)
MM3 = Point(mm3)
ll1 = Segment(AA1, NM11)
ll2 = Segment(AA2, NM22)
ll3 = Segment(AA3, NM33)
rr1 = Segment(NM11, EE1)
rr2 = Segment(NM22, EE2)
rr3 = Segment(NM33, EE3)
Dreieck_{ABC}, c, a, b = Polygon(A, B, C)
g = Line(C, C')
h = Line(B, B')
i = Line(A', A)
j = Segment(C', A)
positiv_C: "\frac{\overline{AC'}}{\overline{C'B}}=" + Verhältnis_c
c_11 = Vector(A, C')
c_21 = Vector(A, B)
negativ_C: "\frac{\overline{AC'}}{\overline{C'B}}=" + (-Verhältnis_c)
a_11 = Vector(B, A')
a_21 = Vector(B, C)
positiv_A: "\frac{\overline{BA'}}{\overline{A'C}}=" + Verhältnis_a
negativ_A: "\frac{\overline{BA'}}{\overline{A'C}}=" + (Verhältnis_a (-1))
positiv_B: "\frac{\overline{CB'}}{\overline{B'A}}=" + Verhältnis_b
negativ_B: "\frac{\overline{CB'}}{\overline{B'A}}=" + (Verhältnis_b (-1))
b_11 = Vector(C, B')
b_21 = Vector(C, A)
Ceva_{positiv}: "\frac{\overline{AC'}}{\overline{C'B}}\cdot \frac{\overline{BA'}}{\overline{A'C}}\cdot \frac{\overline{CB'}}{\overline{B'A}}=" + (Verhältnis_a Verhältnis_b Verhältnis_c)
Ceva_{negativ}: "\frac{\overline{AC'}}{\overline{C'B}}\cdot \frac{\overline{BA'}}{\overline{A'C}}\cdot \frac{\overline{CB'}}{\overline{B'A}}=" + Produkt_{neg}
Ergebnis = If((a_11 a_21 < 0 ∨ a_11 a_21 > a_21 a_21) ∧ (b_11 b_21 < 0 ∨ b_11 b_21 > b_21 b_21) ∧ (c_11 c_21 < 0 ∨ c_11 c_21 > c_21 c_21) ∨ (a_11 a_21 < 0 ∨ a_11 a_21 > a_21 a_21) ∧ b_11 b_21 > 0 ∧ b_11 b_21 < b_21 b_21 ∧ c_11 c_21 > 0 ∧ c_11 c_21 < c_21 c_21 ∨ (b_11 b_21 < 0 ∨ b_11 b_21 > b_21 b_21) ∧ a_11 a_21 > 0 ∧ a_11 a_21 < a_21 a_21 ∧ c_11 c_21 > 0 ∧ c_11 c_21 < c_21 c_21 ∨ (c_11 c_21 < 0 ∨ c_11 c_21 > c_21 c_21) ∧ a_11 a_21 > 0 ∧ a_11 a_21 < a_21 a_21 ∧ b_11 b_21 > 0 ∧ b_11 b_21 < b_21 b_21, Ceva_{negativ}, Ceva_{positiv})
E = Midpoint(a)
M_{c} = Midpoint(c)
A'_1 = Rotate(A, 25°, B)
α = Angle(A, B, A'_1)
C'_1 = Rotate(C, 45°, A)
β = Angle(C, A, C'_1)
D = Midpoint((x(C'_1), y(C'_1) + 0.9), A)
B'_1 = Rotate(B, 45°, C)
γ = Angle(B, C, B'_1)
H_1 = Midpoint(B'_1, C)
w_β = AngularBisector(A, B, C)
w_α = AngularBisector(B, A, C)
I = Intersect(w_β, w_α)
M_{a} = Midpoint(a)
L_1 = Midpoint(b)
m = Segment(A, M_{a})
n = Segment(B, L_1)
S = Intersect(m, n)
p = OrthogonalLine(C, d)
q = OrthogonalLine(B, f)
H = Intersect(q, p)
r = OrthogonalLine(I, c)
s = OrthogonalLine(I, a)
K = Intersect(s, a)
M_1 = Intersect(r, c)
t = Segment(M_1, C)
b_1 = Segment(A, K)
G = Intersect(t, b_1)
M' = Mirror(M_1, M_{c})
X_1 = Mirror(K, M_{a})
c_1 = Segment(M', C)
d_1 = Segment(X_1, A)
N = Intersect(c_1, d_1)
e_1 = LineBisector(c)
f_1 = LineBisector(a)
U = Intersect(e_1, f_1)
M_{b} = Midpoint(b)
g_1 = AngularBisector(M_{a}, M_{c}, M_{b})
h_1 = AngularBisector(M_{c}, M_{b}, M_{a})
Sp = Intersect(g_1, h_1)
M = Midpoint(U, H)
B'_2 = Mirror(B, w_β)
L' = Mirror(L_1, w_β)
n' = Segment(B'_2, L')
F = Mirror(M_{a}, w_α)
k_1 = Circle(C, B)
p_1 = Circle(B, C)
q_1 = Circle(B, A)
r_1 = Circle(A, B)
J, P = Intersect(q_1, r_1)
Q, R = Intersect(p_1, k_1)
i_1 = Segment(R, A)
j_1 = Segment(J, C)
F_1 = Intersect(j_1, i_1)
l_1 = Line(Q, A)
m_1 = Line(P, C)
F_2 = Intersect(m_1, l_1)
n_1, s_1 = AngularBisector(e, d)
t_1, w_γ = AngularBisector(f, e)
X_{ } = Intersect(n_1, t_1)
Z = Intersect(w_γ, n_1)
c_2 = Line(X_{ }, M_{a})
d_2 = Line(Z, M_{c})
Mi = Intersect(c_2, d_2)
e_2, f_2 = AngularBisector(f, d)
Y = Intersect(e_2, t_1)
X_1Inkreismittelpunkt = false
X_3Umkreismittelpunkt = false
X_4Höhenschnittpunkt = false
X_5Feuerbachkreis = false
X_8Nagelpunkt = false
X_9Mittenpunkt = false
X_{10}Spiekerpunkt = false
X_{13,14}Fermatpunkte = false
X_7Gergeoinnepunkt = false
X_xAnkreismittelpunkte = false
X_2Schwerpunkte = false
g_2 = OrthogonalLine(Z, d)
Z_c = Intersect(g_2, d)
h_2 = Circle(Z, Z_c)
O, Z_b = Intersect(h_2, f)
T, Z_a = Intersect(h_2, e)
i_2 = OrthogonalLine(X_{ }, e)
X_a = Intersect(i_2, e)
k_2 = Circle(X_{ }, X_a)
W, X_b = Intersect(k_2, f)
B_1, X_c = Intersect(k_2, d)
j_2 = OrthogonalLine(Y, e)
Y_a = Intersect(j_2, e)
p_2 = Circle(Y, Y_a)
Y_c, C_1 = Intersect(p_2, d)
D_1, Y_b = Intersect(p_2, f)
Ankreisberührpunkte = false
l_2 = OrthogonalLine(I, a)
V = Intersect(e, a)
I_a = Intersect(l_2, a)
q_2 = Circle(I, I_a)
A_1, I_b = Intersect(q_2, b)
G_1, I_c = Intersect(q_2, d)
Inkreisberührpunkte = false
Seitenmitten = false
W_β = Intersect(w_β, f)
W_γ = Intersect(w_γ, d)
W_α = Intersect(w_α, e)
SeitenWinkelhalbierende = false
sh_a = Line(A, M_{a})
sh_b = Line(B, M_{b})
k = Mirror(sh_b, w_β)
l = Mirror(sh_a, w_α)
L = Intersect(k, l)
o = false
h_c = OrthogonalLine(C, d)
h_b = OrthogonalLine(B, f)
h_a = OrthogonalLine(A, e)
H_a = Intersect(h_a, e)
H_b = Intersect(h_b, f)
H_c = Intersect(h_c, d)
u = false
E_C = Midpoint(H, C)
E_A = Midpoint(H, A)
E_B = Midpoint(H, B)
v = false