Pizza, me voilà !

Maintenant comment partager la pizza en
cinq parts avec ce pentagone ?

Stratégie 1

Etapes

On évalue le point T comme indiqué sur le schéma
(T est le point d'intersection de deux droites reliant un sommet
au milieu de ''son côté opposé'').
C'est ce point que l'on va ensuite placer au centre de la pizza
(l'évaluation du centre de la pizz' peut se faire de différente façon...

On place le ''centre'' T du pentagone sur le centre de la pizza.
Puis on marque la pizza en coupant selon les axes reliant
un sommet du pentagone au point T...

Stratégies 2, 3, 4, 5 et 6

Lors de la stratégie 1, on a construit le point T comme intersection
de deux droites reliant un sommet du pentagone au milieu de
''son côté opposé''... mais si on prend deux autres de ces droites,
les parts créées changent-elles ? Autrement dit, toutes ces droites
sont-elles concourantes ?

Eh bien non, elles ne le sont pas ! Il ne reste plus qu'à
reprendre la stratégie 1 pour les autres points d'intersection...

Je veux le faire ! :p

J'veux les résultats directement !

Taux de variation maximal des aires des parts : 

Il vous faut redéfinir le point F_1 : ''clic droit'' sur le point F_1, ''propriétés'', ''Définition :'',
puis modifiez la définition, par exemple : ''Intersection[c_3, a_3]''...
Les taux de variation maximal ci-dessous changera automatiquement.

PARTAGE D'UNE PIZZA EN 5 PARTS

en pliant une feuille de papier au format A4...

(méthode du livre 'Les maths au quotidien')

Johan MATHIEU [20-07-2009]

Déplace ce 'carré magique' pour ne laisser apparaître 
que le pentagone construit en pliant la feuille...

A = (5.153996843919107, 7.9144210199931955)
c = Circle(A, 21.0)
B = Point(c)
a = Segment(A, B)
b = OrthogonalLine(B, a)
d = Circle(B, 29.7)
C, C'_2 = Intersect(d, b)
h = Segment(B, C)
e = Segment(A, C)
C_1 = Midpoint(e)
a_3 = Segment(B, C_1)
D = Mirror(B, C_1)
f = Segment(A, D)
m = Segment(B, D)
i = Segment(B, C_1)
k = OrthogonalLine(C_1, i)
F = Intersect(k, h)
b_1 = Segment(D, F)
E = Intersect(f, k)
s = Segment(E, D)
g_1 = Circle(C_1, 0.5)
H, I = Intersect(g_1, k)
h_1 = Line(I, m)
K = Intersect(b_1, h_1)
j_1 = AngularBisector(F, K, I)
L = Intersect(j_1, k)
i_1 = Line(H, m)
J = Intersect(i_1, s)
k_1 = AngularBisector(E, J, H)
M = Intersect(k_1, k)
f_2 = Ray(J, M)
s_2 = Segment(A, B)
N = Intersect(f_2, s_2)
q_1 = Segment(N, B)
W = Midpoint(q_1)
p_4 = Segment(W, L)
C' = Mirror(C, k)
d_1 = Segment(B, C')
r_1 = Segment(N, M)
U = Midpoint(r_1)
O = Intersect(d_1, j_1)
n_4 = Segment(U, O)
t_1 = Segment(B, O)
V = Midpoint(t_1)
m_4 = Segment(M, V)
a_2 = Segment(O, L)
S = Midpoint(a_2)
c_3 = Segment(S, N)
g = Segment(D, C)
j = Segment(C_1, D)
l = Segment(E, F)
etape = 0 (slider from 0 to 5, step 1)
α = Angle(F, C_1, B)
n = Line(C', C)
G = Intersect(n, k)
β = Angle(C', G, F)
p = Segment(G, C')
q = Segment(C, G)
r = Segment(A, E)
t = Segment(D, C)
c_1 = Segment(F, C')
a_1 = Segment(B, C)
e_1 = Segment(E, F)
γ = Angle(L, K, I)
δ = Angle(F, K, L)
ε = Angle(H, J, M)
ζ = Angle(M, J, E)
N_2 = (5.489233351050635, 30.131682782286585)
C'' = Mirror(C', j_1)
A'' = Mirror(A, k_1)
l_1 = Segment(C'', C')
η = Angle(l_1, j_1)
m_1 = Segment(A, A'')
θ = Angle(m_1, k_1)
F'' = Mirror(F, j_1)
E'' = Mirror(E, k_1)
n_1 = Segment(F, F'')
p_1 = Segment(E, E'')
ι = Angle(n_1, j_1)
κ = Angle(p_1, k_1)
s_1 = Segment(M, L)
b_2 = Segment(O, C'')
c_2 = Segment(L, F'')
d_2 = Segment(F'', C'')
e_2 = Segment(M, E'')
g_2 = Segment(A'', N)
h_2 = Segment(E, B)
i_2 = Segment(A, N)
j_2 = Segment(A, E)
k_2 = Segment(F, C')
l_2 = Segment(C', O)
m_2 = Segment(E, M)
n_2 = Segment(L, F)
P = Point(h_1)
p_2 = OrthogonalLine(P, m)
Q = Intersect(p_2, i_1)
R = Intersect(m, p_2)
q_2 = Segment(Q, R)
r_2 = Segment(R, P)
o = false
poly1, b_3, n_3, m_3, l_3, o_1 = Polygon(B, N, M, L, O)
λ, μ, ν, ξ, ο = Angle(poly1)
f_1 = Segment(A'', E'')
t_2 = Ray(K, L)
u = false
etapess1 = 0 (slider from 0 to 2, step 1)
T = Intersect(c_3, a_3)
d_3 = Circle(T, 15.0)
e_3 = Ray(T, B)
f_3 = Ray(T, M)
g_3 = Ray(T, L)
h_3 = Ray(T, O)
i_3 = Ray(T, N)
Z = Intersect(d_3, i_3)
A_1 = Intersect(d_3, f_3)
B_1 = Intersect(d_3, g_3)
D_1 = Intersect(d_3, h_3)
E_1 = Intersect(d_3, e_3)
j_3 = Segment(T, N)
k_3 = Segment(N, Z)
p_3 = Segment(T, M)
q_3 = Segment(M, A_1)
r_3 = Segment(T, B)
s_3 = Segment(B, E_1)
t_3 = Segment(T, O)
a_4 = Segment(O, D_1)
b_4 = Segment(T, L)
c_4 = Segment(L, B_1)
d_4 = CircleSector(T, D_1, E_1)
e_4 = CircleSector(T, E_1, Z)
f_4 = CircleSector(T, Z, A_1)
g_4 = CircleSector(T, A_1, B_1)
h_4 = CircleSector(T, B_1, D_1)
airee4 = Area(e_4)
Textee4: "Aire = " + airee4
airef4 = Area(f_4)
Textef4: "Aire = " + airef4
aireg4 = Area(g_4)
Texteg4: "Aire = " + aireg4
aireh4 = Area(h_4)
Texteh4: "Aire = " + aireh4
aired4 = Area(d_4)
Texted4: "Aire = " + aired4
i_4 = Segment(L, S)
j_4 = Segment(S, O)
k_4 = Segment(M, C_1)
l_4 = Segment(C_1, L)
v = false
zoom = 0 (slider from 0 to 1, step 1)
w = false
z = false
texte2: "Taux de variation maximal des aires des parts : " + ((Max[{aired4, airee4, airef4, aireg4, aireh4}] - Min[{aired4, airee4, airef4, aireg4, aireh4}]) * 100.0 / Min[{aired4, airee4, airef4, aireg4, aireh4}]) + "%"
F_1 = Intersect(n_4, p_4)
q_4 = Ray(F_1, B)
r_4 = Ray(F_1, O)
s_4 = Ray(F_1, L)
t_4 = Ray(F_1, M)
a_5 = Ray(F_1, N)
G_1 = Intersect(d_3, a_5)
H_1 = Intersect(d_3, q_4)
I_1 = Intersect(d_3, r_4)
J_1 = Intersect(d_3, s_4)
K_1 = Intersect(d_3, t_4)
c_5 = CircleSector(F_1, I_1, H_1)
d_5 = CircleSector(F_1, H_1, G_1)
e_5 = CircleSector(F_1, G_1, K_1)
f_5 = CircleSector(F_1, K_1, J_1)
g_5 = CircleSector(F_1, J_1, I_1)
aired5 = Area(d_5)
Texted5: "Aire = " + aired5
airee5 = Area(e_5)
Textee5: "Aire = " + airee5
airef5 = Area(f_5)
Textef5: "Aire = " + airef5
aireg5 = Area(g_5)
Texteg5: "Aire = " + aireg5
airec5 = Area(c_5)
Textec5: "Aire = " + airec5
texte8: (Max[{airec5, aired5, airee5, airef5, aireg5}] - Min[{airec5, aired5, airee5, airef5, aireg5}]) * 100.0 / Min[{airec5, aired5, airee5, airef5, aireg5}] + "%"
b_5 = Segment(B, F)
L_1 = Intersect(a_1, b_5)
M_1 = Intersect(a_2, t_2)
N_1 = Intersect(a_1, b_5)
O_1 = Intersect(a_1, t_2)
h_5 = Segment(F'', O_1)
P_1 = Intersect(h_2, f_2)
i_5 = Segment(B, O_1)
j_5 = Segment(O_1, C)
k_5 = Segment(P_1, E'')
Q_1 = Intersect(h_2, g_2)
l_5 = Segment(P_1, Q_1)
m_5 = Segment(Q_1, B)
poly2, b_6, e_6, a_6 = Polygon(E, A, B)
poly3, b_7, c', f_6, e_7 = Polygon(B, C', F, E)
poly4, a_7, b_8, c_6, d_6 = Polygon(A, B, C, D)
poly5, l_6, f'', o_2 = Polygon(F'', O_1, L)
poly6, f''_1, c'', o_3, o_4 = Polygon(F'', C'', O, O_1)
poly7, n_5, a'', e'', m_6 = Polygon(N, A'', E'', M)
poly8, m_7, e''_1, a''_1, q_5, b_9, o_5, c''_1, f''_2, l_7 = Polygon(M, E'', A'', Q_1, B, O, C'', F'', L)
poly9, b_{10}, n_6, q_6 = Polygon(N, Q_1, B)
u_1 = Vector(A, A'')
v_1 = Vector(E, E'')
w_1 = Vector(C', C'')
z_1 = Vector(F, F'')
V_1 = (6.309402666469369, -3.529598555742729)
W_1 = Point(Circle[V_1, 25.0])
k_6 = Segment(V_1, W_1)
poly11, p_6, r_6, s_6, t_6, Z_1, A_2 = Polygon(V_1, W_1, 4.0)
p_5 = Segment(V_1, A_2)
r_5 = Segment(A_2, Z_1)
s_5 = Segment(Z_1, W_1)
t_5 = Segment(W_1, V_1)
u_2 = Vector(D, B)