Author: MatheSchmidt

f(x) = 1.0 / 2.0 * (x - 3.0)^(3.0) - 2.0 * (x - 3.0) + 1.0
A1 = 0
A2: A1 + 0.5
A3: A2 + 0.5
A4: A3 + 0.5
A5: A4 + 0.5
A6: A5 + 0.5
A7: A6 + 0.5
A8: A7 + 0.5
A9: A8 + 0.5
A10: A9 + 0.5
A11: A10 + 0.5
A12: A11 + 0.5
B1: f(A1)
B2: f(A2)
B3: f(A3)
B4: f(A4)
B5: f(A5)
B6: f(A6)
B7: f(A7)
B8: f(A8)
B9: f(A9)
B10: f(A10)
B11: f(A11)
B12: f(A12)
P_1: (A1, B1)
P_2: (A2, B2)
P_3: (A3, B3)
P_4: (A4, B4)
P_5: (A5, B5)
P_6: (A6, B6)
P_7: (A7, B7)
P_8: (A8, B8)
P_9: (A9, B9)
P_{10}: (A10, B10)
P_{11}: (A11, B11)
P_{12}: (A12, B12)
Liste1: {P_1, P_2, P_3, P_4, P_5, P_6, P_7, P_8, P_9, P_{10}, P_{11}, P_{12}}
C1 = Circle((A1, B1), 0.2)
C2 = Circle((A2, B2), 0.2)
C3 = Circle((A3, B3), 0.2)
C4 = Circle((A4, B4), 0.2)
C5 = Circle((A5, B5), 0.2)
C6 = Circle((A6, B6), 0.2)
C7 = Circle((A7, B7), 0.2)
C8 = Circle((A8, B8), 0.2)
C9 = Circle((A9, B9), 0.2)
C10 = Circle((A10, B10), 0.2)
C11 = Circle((A11, B11), 0.2)
C12 = Circle((A12, B12), 0.2)
D1 = Point(C1)
D2 = Point(C2)
D3 = Point(C3)
D4 = Point(C4)
D5 = Point(C5)
D6 = Point(C6)
D7 = Point(C7)
D8 = Point(C8)
D9 = Point(C9)
D10 = Point(C10)
D11 = Point(C11)
D12 = Point(C12)
E1 = Line((A1, B1), D1)
E2 = Line((A2, B2), D2)
E3 = Line((A3, B3), D3)
E4 = Line((A4, B4), D4)
E5 = Line((A5, B5), D5)
E6 = Line((A6, B6), D6)
E7 = Line((A7, B7), D7)
E8 = Line((A8, B8), D8)
E9 = Line((A9, B9), D9)
E10 = Line((A10, B10), D10)
E11 = Line((A11, B11), D11)
E12 = Line((A12, B12), D12)
F1, G1 = Intersect(C1, E1)
F2, G2 = Intersect(C2, E2)
F3, G3 = Intersect(C3, E3)
F4, G4 = Intersect(C4, E4)
F5, G5 = Intersect(C5, E5)
F6, G6 = Intersect(C6, E6)
F7, G7 = Intersect(C7, E7)
F8, G8 = Intersect(C8, E8)
F9, G9 = Intersect(C9, E9)
F10, G10 = Intersect(C10, E10)
F11, G11 = Intersect(C11, E11)
F12, G12 = Intersect(C12, E12)
H1 = Segment(D1, G1)
H2 = Segment(D2, G2)
H3 = Segment(D3, G3)
H4 = Segment(D4, G4)
H5 = Segment(D5, G5)
H6 = Segment(D6, G6)
H7 = Segment(D7, G7)
H8 = Segment(D8, G8)
H9 = Segment(D9, G9)
H10 = Segment(D10, G10)
H11 = Segment(D11, G11)
H12 = Segment(D12, G12)
I2: (A2, Slope[E2])
I3: (A3, Slope[E3])
I4: (A4, Slope[E4])
I5: (A5, Slope[E5])
I6: (A6, Slope[E6])
I7: (A7, Slope[E7])
I8: (A8, Slope[E8])
I9: (A9, Slope[E9])
I10: (A10, Slope[E10])
I11: (A11, Slope[E11])
I12: (A12, Slope[E12])

Tool: Ankreise
Gib 3 Punte an
Input: A, B, C
Output: g_a, g_b, g_c, w'_α, w'_β, w'_γ, Inkreis, K_A, K_B, K_C, A', B', C', w_α, w_β, w_γ

g_1 = Line(B, C)
g_a = Line(B, C)
g_2 = Line(C, A)
g_b = Line(C, A)
g_3 = Line(A, B)
g_c = Line(A, B)
w'_1, w_5 = AngularBisector(g_c, g_b)
w'_α, w_1 = AngularBisector(g_c, g_b)
w'_2, w_4 = AngularBisector(g_a, g_b)
w'_γ, w_3 = AngularBisector(g_a, g_b)
w'_3, w_6 = AngularBisector(g_c, g_a)
w'_β, w_2 = AngularBisector(g_c, g_a)
A'_1 = Intersect(w'_β, w'_γ)
A' = Intersect(w'_β, w'_γ)
C'_1 = Intersect(w'_β, w'_α)
C' = Intersect(w'_β, w'_α)
B'_1 = Intersect(w'_γ, w'_α)
B' = Intersect(w'_γ, w'_α)
w_α = Ray(A, A')
w_β = Ray(B, B')
w_γ = Ray(C, C')
a = Segment(B, C)
l_a = OrthogonalLine(A', g_a)
l_b = OrthogonalLine(B', g_b)
l_c = OrthogonalLine(C', g_c)
L_a = Intersect(l_a, g_a)
L_b = Intersect(l_b, g_b)
L_c = Intersect(l_c, g_c)
K_α = Circle(A', L_a)
K_β = Circle(B', L_b)
K_γ = Circle(C', L_c)
K_A: K_α
K_B: K_β
K_C: K_γ
I = Intersect(w_1, w_2)
ha = OrthogonalLine(A, a)
H_a = Intersect(ha, a)
Inkreis = Circle(I, H_a)

Tool: Quadratische Bézierkurve (command Bezier2)
Input: A, B, C
Output: loc1

a = Segment(A, B)
b = Segment(B, C)
D = Point(a)
c = Line(D, b)
d = Segment(A, C)
E = Intersect(c, d)
e = Line(E, a)
F = Intersect(b, e)
f = Segment(D, F)
g = Line(A, f)
G = Intersect(c, g)
h = Segment(B, G)
I = Intersect(f, h)
loc1 = Locus(I, D)

Tool: Eulergerade
Gib 3 Punkte an
Input: A, B, C
Output: Vieleck1, Euler, h_a, h_b, h_c, m_a, m_b, m_c, H, M_a, M_b, M_c, S, U, a, b, c, s_a, s_b, s_c

Vieleck1_1, c_1, a_1, b_1 = Polygon(A, B, C)
Vieleck1, c, a, b = Polygon(A, B, C)
M_a = Midpoint(a)
M_b = Midpoint(b)
M_c = Midpoint(c)
m_a = OrthogonalLine(M_a, a)
m_b = OrthogonalLine(M_b, b)
m_c = OrthogonalLine(M_c, c)
U = Intersect(m_a, m_b)
s_a = Segment(M_a, A)
s_b = Segment(B, M_b)
s_c = Segment(M_c, C)
S = Intersect(s_b, s_a)
h_c = OrthogonalLine(C, c)
h_a = OrthogonalLine(A, a)
h_b = OrthogonalLine(B, b)
H = Intersect(h_a, h_b)
Euler = Line(U, H)

Tool: Feuerbach-Kreis (command Feuerbach)
Gib 3 Punkte an
Input: A, B, C
Output: npc, N, poly1, M_b, M_c, M_a, H, E_1, E_2, E_3, F_1, F_2, F_3, c, a, b

poly1, c, a, b = Polygon(A, B, C)
M_b = Midpoint(b)
M_c = Midpoint(c)
M_a = Midpoint(a)
g = OrthogonalLine(B, b)
h = OrthogonalLine(C, c)
i = OrthogonalLine(A, a)
H = Intersect(i, g)
E_3 = Midpoint(H, C)
E_1 = Midpoint(H, A)
E_2 = Midpoint(H, B)
npc = Circle(M_c, M_a, M_b)
F_2 = Intersect(g, b)
F_3 = Intersect(h, c)
F_1 = Intersect(i, a)
N = Center(npc)

Tool: Höhenschnittpunkt
Gib 3 Punkte ein
Input: A, B, C
Output: H_a, H_b, H, H_c, h_a, h_b, h_c

Vieleck1, c, a, b = Polygon(A, B, C)
hc = OrthogonalLine(C, c)
ha = OrthogonalLine(A, a)
hb = OrthogonalLine(B, b)
AB = Line(A, B)
BC = Line(B, C)
CA = Line(C, A)
H_a = Intersect(ha, BC)
H_b = Intersect(hb, CA)
H_c = Intersect(hc, AB)
h_a = Segment(H_a, A)
h_b = Segment(H_b, B)
h_c = Segment(H_c, C)
H = Intersect(h_a, h_b)

Tool: Inkreis
Gib 3 Punkte an
Input: A, B, C
Output: w_α, w_β, w_γ, Inkreis, D, I, r_I

Vieleck1, c, a, b = Polygon(A, B, C)
w_α = AngularBisector(B, A, C)
w_β = AngularBisector(C, B, A)
w_γ = AngularBisector(A, C, B)
I = Intersect(w_α, w_β)
hilfslinie = OrthogonalLine(I, c)
D = Intersect(hilfslinie, c)
r_I = Segment(D, I)
Inkreis = Circle(I, Distance[I, D])

Tool: ParabelScheitelPunkt
Scheitel, Punkt -> Parabel
Input: S, P
Output: f

f(x) = (y(P) - y(S)) / (x(P) - x(S))^(2.0) * (x - x(S))^(2.0) + y(S)

Tool: Par4P
Parabel durch 2 Punkte und mit 2 Tangenten
Input: A, B, C, D
Output: k, a, b

a = Line(C, B)
b = Line(D, A)
E = Intersect(a, b)
H = Midpoint(A, B)
c = Line(E, H)
d = Line(B, c)
e = Line(A, c)
G = Point(e)
J = Mirror(G, b)
f = Line(J, A)
F_1 = Point(d)
I = Mirror(F_1, a)
g = Line(B, I)
F = Intersect(f, g)
h = Line(F, c)
i = OrthogonalLine(B, h)
K = Intersect(h, i)
L = Intersect(a, h)
S = Midpoint(L, K)
M: S + Vector[F, S]
l = OrthogonalLine(M, h)
k = Parabola(F, l)

Tool: Polynom 3 Punkte (command Polynom3Punkte)
Input: A, B, C
Output: p_2

p_2 = Polynomial((x - x(B)) * (x - x(C)) / (x(A) - x(B)) / (x(A) - x(C)) * y(A) + (x - x(A)) * (x - x(C)) / (x(B) - x(A)) / (x(B) - x(C)) * y(B) + (x - x(A)) * (x - x(B)) / (x(C) - x(A)) / (x(C) - x(B)) * y(C))

Tool: Polynom 4 Punkte (command Polynom4Punkte)
Input: A, B, C, D
Output: p_3

p_3 = Polynomial((x - x(B)) * (x - x(C)) * (x - x(D)) / ((x(A) - x(B)) * (x(A) - x(C)) * (x(A) - x(D))) * y(A) + (x - x(A)) * (x - x(C)) * (x - x(D)) / ((x(B) - x(A)) * (x(B) - x(C)) * (x(B) - x(D))) * y(B) + (x - x(A)) * (x - x(B)) * (x - x(D)) / ((x(C) - x(A)) * (x(C) - x(B)) * (x(C) - x(D))) * y(C) + (x - x(A)) * (x - x(B)) * (x - x(C)) / ((x(D) - x(A)) * (x(D) - x(B)) * (x(D) - x(C))) * y(D))

Tool: Schwerpunkt
Gib 3 Punkte an
Input: A, B, C
Output: s_a, s_b, s_c, S

Vieleck1, c, a, b = Polygon(A, B, C)
M_a = Midpoint(a)
M_b = Midpoint(b)
M_c = Midpoint(c)
s_a = Segment(M_a, A)
s_b = Segment(M_b, B)
s_c = Segment(M_c, C)
S = Intersect(s_a, s_b)

Tool: Tangente an 2 Kreise (command Tang2Kreise)
Tang2Kreise[Kreis k_1, Kreis k_2]
Input: k_1, k_2
Output: i, l, C, d, e, A, r, s, F

r_1 = Radius(k_1)
M_1 = Center(k_1)
M_2 = Center(k_2)
r_2 = Radius(k_2)
a, b = Tangent(M_2, Circle[M_1, r_1 - r_2])
a_1, b_1 = Tangent(M_2, Circle[M_1, r_1 - r_2])
c, d = Tangent(a, k_1)
c_1, d_1 = Tangent(a, k_1)
e, f = Tangent(b, k_2)
A = Intersect(d, e)
g_1, h_1 = Tangent(M_1, Circle[M_2, r_2 - r_1])
g, h = Tangent(M_1, Circle[M_2, r_2 - r_1])
i_1, j_1 = Tangent(h, k_1)
i, j = Tangent(h, k_1)
k_3, l_1 = Tangent(g, k_2)
k, l = Tangent(g, k_2)
C = Intersect(i, l)
m = Line(M_1, M_2)
B = Point(k_2)
n = Segment(B, M_2)
p = Line(M_1, n)
E = Intersect(k_1, p, 2.0)
q = Segment(E, B)
F_1 = Intersect(m, q)
F = Intersect(m, q)
r_3, s_1 = Tangent(F, k_2)
r, s = Tangent(F, k_2)

Tool: Umkreis
Gib 3 Punkte an
Input: A, B, C
Output: m_a, m_b, m_c, Umkreis, U

Vieleck1, c, a, b = Polygon(A, B, C)
M_a = Midpoint(a)
M_b = Midpoint(b)
M_c = Midpoint(c)
m_c = OrthogonalLine(M_c, c)
m_a = OrthogonalLine(M_a, a)
m_b = OrthogonalLine(M_b, b)
U = Intersect(m_a, m_b)
Umkreis = Circle(U, A)