Triangle équilatéral inscrit dans un rectangle avec un sommet commun
Author: ÉRIC TROTOUX
Date: 19 février 2010

Énoncé de l'exercice Omega 8 -- Un triangle  équilatéral ABC est inscrit dans le rectangle AIHG. Montrez que la somme des aires des
triangles AIB et AGC est  égale à l'aire du triangle BCH.

En voici une preuve visuelle. On peut faire bouger le point G et les points A et C pour constater la propriété annoncée dans les différentes configurations du rectangle AIHG ainsi obtenues.

A = (-11.289525443876054, -9.049570804300702)
C = (7.751749279674767, -3.2333165775827397)
C_2 = Circle(C, A)
C_1 = Circle(A, C)
B = Intersect(C_2, C_1, 2)
P, c, a, b = Polygon(A, C, B)
D = Midpoint(A, B)
F = Midpoint(B, C)
E = Midpoint(C, A)
e = Circle(F, B)
f = Circle(E, C)
g = Circle(D, B)
G = Point(f)
h = Line(G, C)
i = Line(G, A)
H = Intersect(e, h, 2)
j = Line(H, B)
I = Intersect(g, j, 2)
k = Line(I, A)
T_1 = Area(B, C, H)
T_2 = Area(A, G, C)
T_3 = Area(A, I, B)
T_4: T_2 + T_3
l = Segment(I, H)
m = Segment(H, G)
n = Segment(G, A)
p = Segment(A, I)
q = Segment(C, D)
r = Segment(D, H)
s = Segment(D, G)
t = Segment(B, E)
a_4 = Segment(I, B)
a_3 = Segment(B, H)
a_2 = Segment(H, C)
a_1 = Segment(C, G)
J = Mirror(D, F)
K = Mirror(E, F)
d = Segment(D, J)
a1 = Segment(E, K)
b1 = Segment(K, H)
c1 = Segment(H, J)
d1 = Segment(E, H)
poly1, l_1, e_1, h_1 = Polygon(E, H, K)
poly2, k_1, d_1, h_2 = Polygon(D, H, J)
e1 = Segment(F, H)
f1 = Rotate(e, 59.99999999999999°, E)
L = Mirror(F, D)
g1 = Segment(L, D)
M = Mirror(E, D)
N = Midpoint(L, D)
O = Midpoint(D, I)
Q = Midpoint(I, B)
R = Mirror(Q, O)
h1 = Segment(D, Q)
i1 = Segment(I, D)
j1 = Line(N, O)
S = Intersect(h1, j1)
k1 = Segment(M, S)
T = Intersect(p, k1)
U = Intersect(a_4, j1)
V = Midpoint(T, U)
W = Mirror(S, V)
l1 = Segment(T, W)
Z = Intersect(i1, l1)
m1 = Segment(T, Z)
n1 = Segment(O, S)
p1 = Line(R, g1)
A1 = Intersect(p1, i1)
poly3, a1_1, d_2, r_1 = Polygon(D, R, A1)
poly4, s_1, o, d_3 = Polygon(O, D, S)
poly5, o_1, s_2, q_1, i_1 = Polygon(O, S, Q, I)
poly6, t_1, z, i_2 = Polygon(Z, I, T)
poly7, r_2, t_2, z_1, a1_2 = Polygon(R, T, Z, A1)
B1 = Midpoint(H, K)
C1 = Midpoint(K, F)
D1 = Mirror(B1, C1)
q1 = Line(D1, B1)
E1 = Intersect(q1, a)
r1 = Segment(B1, E1)
F1 = Midpoint(H, B)
G1 = Intersect(r1, a_3)
H1 = Midpoint(B, F)
I1 = Mirror(E1, H1)
J1 = Mirror(G1, F1)
s1 = Segment(I1, J1)
K1 = Midpoint(F, H)
L1 = Rotate(S, 300.00000000000006°, E)
u = Vector(L1, K1)
M1 = Translate(F1, u)
N1 = Mirror(M1, F1)
O1 = Rotate(A1, 300.00000000000006°, E)
P1 = Rotate(R, 300.00000000000006°, E)
t1 = Line(O1, P1)
Q1 = Intersect(t1, s1)
poly8, q1_1, h1_1, i1_1 = Polygon(H1, I1, Q1)
poly9, j1_1, q1_2, m1_1 = Polygon(Q1, M1, J1)
poly10, f1_1, m1_2, q1_3, h1_2 = Polygon(F1, M1, Q1, H1)
poly11, h1_3, n1_1, f1_2 = Polygon(N1, F1, H1)
poly12, g1_1, n1_2, h1_4, e1_1 = Polygon(G1, N1, H1, E1)
R1 = Midpoint(C, G)
a2 = Segment(E, R1)
b2 = Segment(B1, F)
S1 = Midpoint(E, H)
T1 = Intersect(a_3, b2)
U1 = Mirror(T1, F1)
v = Vector(B1, H)
V1 = Translate(T1, v)
W1 = Translate(G1, v)
poly13, q1_4, b_1, i1_2 = Polygon(B, I1, Q1)
poly14, b_2, u1, q1_5 = Polygon(U1, Q1, B)
poly15, e1_2, f_1, t1_1, g1_2 = Polygon(E1, F, T1, G1)
poly16, f_2, t1_2, h_3 = Polygon(T1, H, F)
Z1 = Midpoint(A, G)
c2 = Segment(E, Z1)
d2 = Segment(G, E)
e2 = Segment(G, F)
A2 = Intersect(a2, e2)
w = Vector(R1, G)
B2 = Translate(A2, w)
C2 = Midpoint(A2, B2)
D2 = Mirror(G, C2)
f2 = Segment(D2, B2)
E2 = Intersect(d2, f2)
F2 = Translate(F, w)
g2 = Segment(F2, Z1)
G2 = Intersect(g2, d2)
poly17, g2_1, z1, e_2 = Polygon(Z1, E, G2)
poly18, g2_2, e2_1, b2_1, z1_1 = Polygon(G2, E2, B2, Z1)
poly19, b2_2, e2_2, g_1 = Polygon(E2, G, B2)
poly20, g_2, e_3, a2_1 = Polygon(E, A2, G)
poly21, g_3, a2_2, r1_1 = Polygon(A2, R1, G)
poly22, j1_2, q1_6, u1_1 = Polygon(Q1, U1, J1)
H2 = Midpoint(I, L)
I2 = Midpoint(M, I)
v1 = Vector(I2, I)
J2 = Translate(H2, v1)
K2 = Translate(D, v1)
h2 = Segment(J2, K2)
L2 = Midpoint(D, F)
i2 = Segment(O, L2)
M2 = Intersect(p, r_2)
N2 = Intersect(p, h2)
O2 = Intersect(i1, h2)
P2 = Intersect(h1, i2)
j2 = Segment(O, P2)
k2 = Segment(N2, O2)
l2 = Segment(F, F1)
m2 = OrthogonalLine(Z1, e2)
Q2 = Intersect(m2, d2)
n2 = Segment(Z1, Q2)
poly23, c_1, a2_3, r1_2 = Polygon(A2, R1, C)
poly24, c_2, a2_4, e_4 = Polygon(A2, E, C)
R2 = Mirror(B2, c2)
S2 = Mirror(E2, c2)
T2 = Mirror(Q2, c2)
U2 = Mirror(G2, c2)
poly25, s2, a_5, r2 = Polygon(A, R2, S2)
poly26, e_5, z1_2, u2 = Polygon(Z1, U2, E)
poly27, p2, q2, t2 = Polygon(Z1, U2, T2)
p2 = Segment(Z1, T2)
V2 = Mirror(O, h1)
poly28, q_2, s_3, v2, b_3 = Polygon(Q, S, V2, B)
poly29, q2, t2, a3 = Polygon(D, V2, P2)
W2 = Mirror(T, a1_1)
Z2 = Mirror(N2, a1_1)
A3 = Mirror(Z, a1_1)
B3 = Mirror(O2, a1_1)
C3 = Mirror(A1, a1_1)
q2 = Segment(P2, V2)
poly30, d_5, r_3, c3 = Polygon(R, C3, D)
poly31, b3, d3, e3, f3 = Polygon(B3, Z2, R, C3)
poly32, a3_1, a_6, w2_1 = Polygon(A, W2, A3)
t2 = Segment(Z2, B3)
D3 = Mirror(F, a_3)
b3 = Segment(B, D3)
d3 = Segment(H, D3)
E3 = Midpoint(B, D3)
F3 = Mirror(I1, a_3)
G3 = Mirror(E1, a_3)
H3 = Mirror(Q1, a_3)
w1 = Vector(D3, H)
I3 = Translate(G3, w1)
J3 = Translate(E3, w1)
K3 = Translate(F3, w1)
L3 = Translate(H3, w1)
M3 = Translate(U1, w1)
N3 = Translate(J1, w1)
O3 = Translate(M1, w1)
P3 = Translate(F1, w1)
poly33, l3, f_3, m3 = Polygon(F, M3, L3)
poly34, l3_1, m3_1, n3 = Polygon(M3, N3, L3)
poly35, k3, f_4, l3_2 = Polygon(F, L3, K3)
poly36, n3_1, o3, l3_3 = Polygon(O3, L3, N3)
poly37, k3_1, l3_4, j3 = Polygon(L3, J3, K3)
poly38, j3_1, p3, o3_1, l3_5 = Polygon(J3, P3, O3, L3)
Q3 = Intersect(f1_2, n1_2)
R3 = Intersect(l2, n1_2)
S3 = Intersect(g1_2, l2)
T3 = Mirror(S3, a_3)
U3 = Mirror(R3, a_3)
V3 = Translate(U3, w1)
W3 = Translate(T3, w1)
poly39, j3_2, p3_1, v3 = Polygon(P3, V3, J3)
poly40, h_4, w3, i3 = Polygon(W3, I3, H)
poly41, j3_3, i3_1, w3_1, v3_1 = Polygon(J3, I3, W3, V3)
poly42, c_3, f_5, m3_2 = Polygon(F, M3, C)
Z3 = Mirror(W3, P3)
g3 = Segment(N3, Z3)
h3 = Line(L3, O3)
A4 = Intersect(a_2, h3)
q3 = Segment(A4, O3)
poly43, p2_1, v2_1, s_4 = Polygon(V2, S, P2)
poly44, a4, o3_2, p3_2 = Polygon(O3, P3, A4)
poly45, z2, b3_1, a3_2, w2 = Polygon(Z2, B3, A3, W2)
poly46, z3, a4_1, o3_3, n3_2 = Polygon(Z3, A4, O3, N3)
poly47, r2_1, s2_1, t2_1, z1_3 = Polygon(R2, S2, T2, Z1)
poly48, m3_3, n3_3, z3_1, c_4 = Polygon(M3, N3, Z3, C)