Animation

Pour conjecturer sur le périmètre d'un cercle

On peut déplacer le point A pour faire varier le diamètre du cercle.
Observe l'animation et après quelques essais émets une conjecture.

t = -26.5 (slider from -100 to 250, step 0.1)
A = (2.6127740736354386, 0.6382364994899731)
O = Intersect(xAxis, yAxis)
c = Circle(O, A)
E, F = Intersect(c, xAxis)
C, D = Intersect(c, yAxis)
G = Point(D, Vector[3.141592653589793 * Vector[E, F]])
v = Vector(D, G)
B = Mirror(A, O)
a = Segment(A, B)
b = Ray(D, Vector[E, F])
α = Angle(F, O, A)
G_1 = Point(yAxis)
H1 = Point(yAxis)
u = Vector(G_1, H1)
H = Point(O, Vector[(t - 100) * v / 50])
e = Circle(H, a / 2)
I, L = Intersect(e, b)
f = Segment(D, I)
I' = Rotate(I, 2 * 3.141592653589793 * (3.141592653589793 * a - f) / (3.141592653589793 * a), H)
g = CircleArc(H, I, I')
h = Segment(H, I')
A'_1 = Rotate(I, 2 * 3.141592653589793 * (3.141592653589793 * a - f) / (3.141592653589793 * a) - 1.5 * 3.141592653589793 + α, H)
A'' = Mirror(A'_1, H)
i = Segment(A'', A'_1)
k = Circle(O, A)
j = Segment(A, B)
A'_2 = Rotate(A, (-(t * α)) / 20, O)
B'_1 = Rotate(B, (-(t * α)) / 20, O)
a'_1 = Segment(A'_2, B'_1)
l = Segment(E, F)
m = Segment(A, B)
c' = Translate(c, v)
A' = Translate(A, v)
B' = Translate(B, v)
diamètre = Segment(A', B')
périmètre = Segment(D, Point[D, v])
E'_1 = Translate(E, Vector[(t - 20) * Vector[E, D] / 20])
F'_1 = Translate(F, Vector[(t - 20) * Vector[E, D] / 20])
l'_1 = Segment(E'_1, F'_1)
E'_2 = Translate(E, Vector[E, D])
F'_2 = Translate(F, Vector[E, D])
l'_2 = Segment(E'_2, F'_2)
E'_3 = Translate(E, Vector[E, F'_2])
F'_3 = Translate(F, Vector[E, F'_2])
l'_3 = Segment(E'_3, F'_3)
E'_4 = Translate(E, Vector[(t - 40) * Vector[E, F'_2] / 20])
F'_4 = Translate(F, Vector[(t - 40) * Vector[E, F'_2] / 20])
l'_4 = Segment(E'_4, F'_4)
E'_5 = Translate(E, Vector[(t - 60) * Vector[E, F'_3] / 20])
F'_5 = Translate(F, Vector[(t - 60) * Vector[E, F'_3] / 20])
l'_5 = Segment(E'_5, F'_5)
E'_6 = Translate(E, Vector[E, F'_3])
F'_6 = Translate(F, Vector[E, F'_3])
l'_6 = Segment(E'_6, F'_6)
E' = Translate(E, Vector[(t - 80) * Vector[E, F'_6] / 20])
F' = Translate(F, Vector[(t - 80) * Vector[E, F'_6] / 20])
l' = Segment(E', F')
d = Segment(Point[D, Vector[E, F]], Point[D, Vector[E, F] + u])
liste2 = Sequence(Segment[Point[D, Vector[k * Vector[E, F]]], Point[D, Vector[k * Vector[E, F] + 0.5 * u]]], k, 0, 0.9, 0.1)
n = Segment(D, Point[D, u])
texte2 = Text("0", D - 2 * u)
texte3 = Text("1  diamètre", D + 0.9 * Vector[E, F] - 2 * u)
texte4 = Text("2  diamètres", D + 1.9 * Vector[E, F] - 2 * u)
liste1 = Sequence(Segment[Point[D, Vector[k * Vector[E, F]]], Point[D, Vector[k * Vector[E, F] + 0.5 * u]]], k, 1.1, 1.9, 0.1)
p = Segment(Point[D, Vector[2 * Vector[E, F]]], Point[D, Vector[2 * Vector[E, F] + u]])
q = Segment(Point[D, Vector[3 * Vector[E, F]]], Point[D, Vector[3 * Vector[E, F] + u]])
liste3 = Sequence(Segment[Point[D, Vector[k * Vector[E, F]]], Point[D, Vector[k * Vector[E, F] + 0.5 * u]]], k, 2.1, 2.9, 0.1)
texte5 = Text("3  diamètres", D + 2.9 * Vector[E, F] - 2 * u)
texte6 = Text("4  diamètres", D + 3.9 * Vector[E, F] - 2 * u)
liste4 = Sequence(Segment[Point[D, Vector[k * Vector[E, F]]], Point[D, Vector[k * Vector[E, F] + 0.5 * u]]], k, 3.1, 3.9, 0.1)
r = Segment(Point[D, Vector[4 * Vector[E, F]]], Point[D, Vector[4 * Vector[E, F] + u]])
J = (7.5, 3.02)
K = (8.88, 3.02)