Si[(t ≥ 2) ∧ (t ≤ 3), y(A) + (y(J) - y(A)) (t - 2), Si[t < 2, y(C) + sin(θ) (x(D) - x(C)) + cos(θ) (y(D) - y(C)), y(J)]]

Si[(t ≥ 2) ∧ (t ≤ 3), x(A) + (x(J) - x(A)) (t - 2), Si[t < 2, x(B) + cos(θ) (x(I) - x(B)) - sin(-θ) (y(I) - y(B)), x(J)]]

y(C) + sin(θ) (x(B) - x(C)) + cos(θ) (y(B) - y(C))

Si[(t ≥ 3) ∧ (t ≤ 4), y(A) + (y(K) - y(A)) (t - 3), Si[t < 2, y(C) + sin(θ) (x(D) - x(C)) + cos(θ) (y(D) - y(C)), y(K)]]

Si[(t ≥ 2) ∧ (t ≤ 3), y(A) + (y(J) -y(A)) (t - 2), Si[t < 2, y(D), y(G)]]

Si[(t ≥ 2) ∧ (t ≤ 3), x(A) + (x(J) - x(A)) (t - 2),   Si[(t < 2), x(D),x(G)   ]]

Si[t ≤ 1, y(A) ,y2]

A

B

C

x(A) + (-x(A) + x(K)) (t - 3)

Si[t ≤ 1, x(A) + t (x(B) - x(A)), x(C) + cos(θ) (x(B) - x(C)) - sin(θ) (y(B) - y(C))]

AC²

AB²

AC²

AB²

BC²

+

=

Lorsqu'un sommet se déplace parallélement au côté opposé,
 son aire ne change pas.

Lorsqu'un triangle tourne autour d'un point,
 son aire ne change pas.

On procède de la même façon avec ce triangle ....

On obtient donc :

Laurent.costa@ac-lyon.fr

\frac{AB²}{2}

\frac{AB²}{2}

Aire  \; (triangle) = \frac{AC²}{2}

K_1 = (-5.56, -6.14)
A_2 = (2, 0)
B_2 = (6, 0)
t = -0.02 (slider from -0.02 to 10, step 1.0E-5)
C_2 = (2, 2)
poly3, c_2, b_1, f, g, F, G = Polygon(C_2, B_2, 4)
m = OrthogonalLine(A_2, c_2)
K = Intersect(m, f)
y10 = If((t ≥ 7) ∧ (t ≤ 8), y(A_2) + (y(K) - y(A_2)) * (t - 7), If[t < 7, y(A_2), y(K)])
x10 = If((t ≥ 7) ∧ (t ≤ 8), x(A_2) + (x(K) - x(A_2)) * (t - 7), If[t < 7, x(A_2), x(K)])
poly4, b_2, a_2, h, i, H, I = Polygon(B_2, A_2, 4)
y9 = If((t ≥ 7) ∧ (t ≤ 8), y(I) + (y(F) - y(I)) * (t - 7), If[t < 7, y(I), y(F)])
x9 = If((t ≥ 7) ∧ (t ≤ 8), x(I) + (x(F) - x(I)) * (t - 7), If[t < 7, x(I), x(F)])
J = Intersect(m, c_2)
y8 = If((t ≥ 7) ∧ (t ≤ 8), y(H) + (y(J) - y(H)) * (t - 7), If[t < 7, y(H), y(J)])
x8 = If((t ≥ 7) ∧ (t ≤ 8), x(H) + (x(J) - x(H)) * (t - 7), If[t < 7, x(H), x(J)])
α = If((t ≥ 5) ∧ (t ≤ 6), (90 * (t - 5))°, If[t < 5, 0, 90*°])
θ = If((t ≥ 1) ∧ (t ≤ 2), (90 * (t - 1))°, If[t < 1, 0, 90*°])
y7 = If((t ≥ 6) ∧ (t ≤ 7), y(A_2) + (y(J) - y(A_2)) * (t - 6), If[t < 6, y(B_2) + sin(α) * (x(I) - x(B_2)) + cos(α) * (y(I) - y(B_2)), y(J)])
x7 = If((t ≥ 6) ∧ (t ≤ 7), x(A_2) + (x(J) - x(A_2)) * (t - 6), If[t < 6, x(B_2) + cos(α) * (x(I) - x(B_2)) - sin(-α) * (y(I) - y(B_2)), x(J)])
y6 = If((t ≤ 5) ∧ (t ≥ 4), y(A_2) + (t - 4) * (y(C_2) - y(A_2)), y(B_2) + cos(α) * (y(C_2) - y(B_2)) - sin(α) * (x(C_2) - x(B_2)))
poly2, a_1, c_1, d, e, D, E = Polygon(A_2, C_2, 4)
y5 = If((t ≥ 3) ∧ (t ≤ 4), y(E) + (y(J) - y(E)) * (t - 3), If[t < 3, y(E), y(J)])
x5 = If((t ≥ 3) ∧ (t ≤ 4), x(E) + (x(J) - x(E)) * (t - 3), If[t < 3, x(E), x(J)])
y4 = If((t ≥ 3) ∧ (t ≤ 4), y(D) + (y(G) - y(D)) * (t - 3), If[t < 3, y(D), y(G)])
x4 = If((t ≥ 3) ∧ (t ≤ 4), x(D) + (x(G) - x(D)) * (t - 3), If[t < 3, x(D), x(G)])
y3 = If((t ≥ 3) ∧ (t ≤ 4), y(A_2) + (y(K) - y(A_2)) * (t - 3), If[t < 3, y(A_2), y(K)])
y2 = If((t ≥ 2) ∧ (t ≤ 3), y(A_2) + (y(J) - y(A_2)) * (t - 2), If[t < 2, y(C_2) + sin(θ) * (x(D) - x(C_2)) + cos(θ) * (y(D) - y(C_2)), y(J)])
x2 = If((t ≥ 2) ∧ (t ≤ 3), x(A_2) + (x(J) - x(A_2)) * (t - 2), If[t < 2, x(C_2) + cos(θ) * (x(D) - x(C_2)) - sin(θ) * (y(D) - y(C_2)), x(J)])
J3 = (2, 2)
y1 = If((t ≤ 1) ∧ (t ≥ 0), y(A_2), y(C_2) + sin(θ) * (x(B_2) - x(C_2)) + cos(θ) * (y(B_2) - y(C_2)))
x1 = If((t ≤ 1) ∧ (t ≥ 0), x(A_2) + t * (x(B_2) - x(A_2)), x(C_2) + cos(θ) * (x(B_2) - x(C_2)) - sin(θ) * (y(B_2) - y(C_2)))
J1: (x1, y1)
poly1, b, c, a = Polygon(C_2, A_2, B_2)
J2: (x2, y2)
poly5, j, k, l = Polygon(J2, J1, J3)
r1 = Distance(B_2, C_2)
x3 = If((t ≥ 3) ∧ (t ≤ 4), x(A_2) + (x(K) - x(A_2)) * (t - 3), If[t < 3, x(A_2), x(K)])
K2: (x3, y3)
K1: (x4, y4)
K3: (x5, y5)
poly6, k2, k1, k3 = Polygon(K1, K3, K2)
L3: (x(B_2), y(B_2))
x6 = If((t ≤ 5) ∧ (t ≥ 4), x(A_2) + (t - 4) * (x(C_2) - x(A_2)), If[t ≥ 5, x(B_2) + cos(-α) * (x(C_2) - x(B_2)) - sin(-α) * (y(C_2) - y(B_2))])
L2: (x6, y6)
L1: (x7, y7)
poly7, n, l_1, m_1 = Polygon(L1, L2, L3)
M: (x8, y8)
N: (x9, y9)
L: (x10, y10)
poly8, n_1, l_2, m_2 = Polygon(L, M, N)
p = Line(C_2, D)
O = Point(p)
P = Point(p)
q = Line(A_2, B_2)
Q: (x(O), y(A_2))
R: (x(P), y(A_2))
r = Segment(O, P)
s = Segment(Q, R)
d_1 = Line(A_2, J)
e_1 = Line(C_2, d_1)
S = Point(e_1)
T = Point(e_1)
f_1 = OrthogonalLine(T, e_1)
g_1 = OrthogonalLine(S, e_1)
U = Intersect(d_1, g_1)
V = Intersect(d_1, f_1)
h_1 = Segment(S, T)
i_1 = Segment(U, V)
Z: (x(A_2), y(H) - 1)
A_1: (x(B_2), y(C_2) + 4)
W: (x(A_2), y(C_2) + 4)
B_1: (x(B_2), y(H) - 1)
j_1 = Segment(W, Z)
k_1 = Segment(B_1, A_1)
p_1 = Segment(U, V)
q_1 = Line(B_2, p_1)
r_1 = OrthogonalLine(U, q_1)
s_1 = OrthogonalLine(V, p_1)
C_1 = Intersect(q_1, r_1)
D_1 = Intersect(q_1, s_1)
t_1 = Segment(C_1, D_1)
β = Angle(B_2, A_2, C_2)
tt: floor(t)
vitesse = 0.09385331168177564
E_1 = Midpoint(n)
F_1 = Midpoint(E_1, B_2)
G_1 = Midpoint(l_2)
H_1 = Midpoint(L, G_1)
I_1: (-4.8, y(K_1))
I_2: (x(I_1), y(I_1) + 0.8)
I_3: (x(I_1) + 0.8, y(I_1))
K_2: (x(K_1) + 0.8, y(K_1))
K_3: (x(K_1), y(K_1) + 0.8)
J_1: (-3.9, y(K_1))
J_3: (x(J_1), y(J_1) + 0.8)
J_2: (x(J_1) + 1.3, y(J_1))
L_1: (-2.4, y(K_1))
L_2: (x(L_1) + 1.3, y(L_1))
L_3: (x(L_1), y(L_1) + 0.8)
M_1: (-1.1, y(K_1))
M_3: (x(M_1), y(M_1) + 0.8)
M_2: (x(M_1) + 1, y(M_1))
N_1: (0.1, y(K_1))
O_1: (2.1, y(K_1))
N_3: (x(N_1), y(N_1) + 0.8)
O_3: (x(O_1), y(O_1) + 0.8)
O_2: (x(O_1) + 1.7, y(O_1))
N_2: (x(N_1) + 1.7, y(N_1))