M \left( x_C+r.\cos\alpha,y_C+r'.\cos(3\alpha)\right)

α

-60°

\text{Le cercle vert a pour 
rayon }r= x'_2-x'_1=\dfrac{2\sqrt\Delta}{3|a|}

\text{Le cercle bleu  a pour 
rayon }r'=\dfrac{ |f(x_1)-f(x_2)|}2=\dfrac{2\sqrt\Delta^3}{27a^2}=\dfrac{|a|r^3}{4}

x_1

x_2

x_3

x_4

x_5

3α

60°

x_1

x_2

x'_2

x_3

x'_1

S_2

S_1

C

A_3

A_2

A_1

v: 1
u: 2
n: u^(2) + (3 * v^(2))
p: (2 * u^(2)) - (6 * v^(2))
q: ((12 * u) * v)
xC = 10 (slider from -25 to 25, step 1)
xc: xC
x1: xc + p
x2: xc + ((-p) + q) / 2
x3: xc + ((-p) - q) / 2
σ1: x1 + x2 + x3
σ2: (x1 * x2) + (x2 * x3) + (x3 * x1)
σ3: ((x1 * x2) * x3)
yC: 2
C: (xc, yC)
d = GCD(GCD[n, p], q)
x'1: xc - n / d
x'2: xc + n / d
g(x) = (((x - x1) * (x - x2)) * (x - x3))
k: g(xc)
a: yC / k
f(x) = (a * g(x))
S1: (x'1, f(x'1))
S2: (x'2, f(x'2))
Δ: σ1^(2) - (3 * σ2)
r: (2 / 3 * sqrt(abs(Δ)))
r': (a * r^(3) / 4)
α0: acos(((-yC)) / abs(r')) / 3
α = 2.478367537831948 rad (slider from 0° to 180°, step 1°)
Q: (xc, yC + (sgn(a) * r'))
P: (xc + r, yC)
M' = Rotate(P, α, C)
M'' = Rotate(Q, ((3 * sgn(a)) * α), C)
M: (x(M'), y(M''))
β = Angle(P, C, M')
d3: x = xc
m_2 = Segment(C, M'')
d2: x = x'2
d1: x = x'1
A = Rotate(P, 120°, C)
E = Rotate(P, 60°, C)
n_2 = Segment(C, A)
p_2 = Segment(C, E)
ε = Angle(P, C, E)
X: (xc - r, yC - r')
Y: (xc + r, yC - r')
Z: (xc + r, yC + r')
W: (xc - r, yC + r')
q1, t, m_1, n_1, p_1 = Polygon(W, X, Y, Z)
A2 = Rotate(P, α0, C)
A3 = Rotate(A2, (2 * π / 3), C)
A1 = Rotate(A3, (2 * π / 3), C)
j_1 = Segment(C, A2)
k_1 = Segment(C, A3)
l_1 = Segment(C, A1)
h_1 = Segment(C, M')
R: (xc + (r * cos(α)), 0)
i = Segment(M', R)
q_1 = Segment(M, M'')
r_1 = Segment(M, R)
δ = Angle(A2, C, A3)
η = Angle(A3, C, A1)
e_1 = Circle(C, r')
X_1: (x1, 0)
X_2: (x2, 0)
X_3: (x3, 0)
X_4: (x'1, 0)
X_5: (x'2, 0)
c = Circle(C, r)
γ = Angle(Q, C, M'')
ζ = Angle(E, C, A)
j = Segment(C, P)
B = Intersect(f, yAxis)
τ1: x'1 + x'2
τ2: (x'1 * x'2)
texte8: "(x'_1,x'_2)=(" + (LaTeX[x'2]) + "," + (LaTeX[x'1]) + ")"
texte3: "(x_1,x_2,x_3)=(" + (LaTeX[x2]) + "," + (LaTeX[x3]) + "," + (LaTeX[x1]) + ")"
h = -3.5 (slider from -10 to 10, step 0.1)
s = Segment(A3, X_3)
l = Segment(A1, X_1)
m = Segment(X_2, A2)
d' = GCD(yC, k)
y'C: yC / d'
k': k / d'
texte1: "a=\frac{" + (LaTeX[y'C]) + "}{" + (LaTeX[k']) + "}"
texte4: "f(x)=\frac{" + (LaTeX[y'C]) + "}{" + (LaTeX[k']) + "}\left(x^3-" + (LaTeX[σ1]) + "x^2+" + (LaTeX[σ2]) + "x-" + (LaTeX[σ3]) + "\right)"
texte8_1: "f'(x)=3a\left(x^2-" + (LaTeX[τ1]) + "x+" + (LaTeX[τ2]) + "\right)"