Le \;point \;de\; Vecten\;\;\;\;\left(  Nombres\;complexes \right) 

Conjecture:\
les\; droites\; (RB),\;(QA)\; et\; (PC)\; étant \;les\; supports \;des\
 hauteurs\; du\; triangle\;PQR\;alors\; elles\; sont\; toujours\ 
concourantes:\;leur\; point \;de\; concours\; est\; l'orthocentre \
du \;triangle\;PQR

R

Q

B

C

P

A

R

Q

P

On a:

 alors (RB) et (PQ) sont perpendiculaires, par suite
    (RB) est la hauteur issue de R du triangle PQR

 alors (PC) et (RQ) sont perpendiculaires, par suite
    (PC) est la hauteur issue de P du triangle PQR

 alors (QA) et (RP) sont perpendiculaires, par suite
    (QA) est la hauteur issue de Q du triangle PQR

A = (-0.8288526077097493, -0.011056689342405696)
B = (4, 0)
C = (2.2453495923640867, -2.9970561873121357)
poly1, c, a, b = Polygon(A, B, C)
poly2, d, e, f, g, D, E = Polygon(A, C, 4)
poly3, h, i, j, k, F, G = Polygon(B, A, 4)
poly4, l, m, n, p, H, I = Polygon(C, B, 4)
R = Midpoint(E, C)
Q = Midpoint(B, I)
P = Midpoint(F, B)
q = Segment(C, P)
r = Segment(Q, A)
s = Segment(B, R)
poly5, p_1, r_1, q_1 = Polygon(R, Q, P)
u = Vector(R, B)
A_1 = (-5.04, 9.82)
B_1 = (-0.54, 10.96)
C_1 = (-2.6, 13.1)
A_2 = (-5.64, 8.84)
B_2 = (-1.14, 9.98)
C_2 = (-3.2, 12.12)
A_3 = (-6.42, 4.2)
B_3 = (-1.92, 5.34)
C_3 = (-3.98, 7.48)
A_4 = (-6.78, 6.3)
B_4 = (-2.28, 7.44)
C_4 = (-4.34, 9.58)
u' = Vector(P, Q)
su: u * u'
v' = Vector(P, R)
v = Vector(Q, A)
w = Vector(P, C)
w' = Vector(Q, R)
sv: v * v'
sw: w * w'
A_5 = (22.84959637188209, 3.9618321995464902)
B_5 = (28.791501133786852, 6.581496598639459)
C_5 = (26.73150113378685, 8.721496598639462)
poly1_1, c_1, a_1, b_1 = Polygon(A_5, B_5, C_5)
poly2_1, d_1, e_1, f_1, g_1, D_1, E_1 = Polygon(A_5, C_5, 4)
poly3_1, h_1, i_1, j_1, k_1, F_1, G_1 = Polygon(B_5, A_5, 4)
R_1 = Midpoint(E_1, C_5)
P_1 = Midpoint(F_1, B_5)
q_2 = Segment(C_5, P_1)
s_1 = Segment(B_5, R_1)
u_1 = Vector(R_1, B_5)
v'_1 = Vector(P_1, R_1)
w_1 = Vector(P_1, C_5)
A_6 = (21.811356009070302, 0.19519274376417384)
B_6 = (27.753260770975054, 2.814857142857146)
C_6 = (25.69326077097506, 4.954857142857147)
poly1_2, c_2, a_2, b_2 = Polygon(A_6, B_6, C_6)
poly2_2, d_2, e_2, f_2, g_2, D_2, E_2 = Polygon(A_6, C_6, 4)
poly3_2, h_2, i_2, j_2, k_2, F_2, G_2 = Polygon(B_6, A_6, 4)
poly4_2, l_2, m_2, n_2, p_5, H_2, I_2 = Polygon(C_6, B_6, 4)
R_2 = Midpoint(E_2, C_6)
Q_2 = Midpoint(B_6, I_2)
P_2 = Midpoint(F_2, B_6)
q_4 = Segment(C_6, P_2)
r_5 = Segment(Q_2, A_6)
s_2 = Segment(B_6, R_2)
poly5_2, p_4, r_4, q_5 = Polygon(R_2, Q_2, P_2)
u_2 = Vector(R_2, B_6)
u'_2 = Vector(P_2, Q_2)
su_2: u_2 * u'_2
v'_2 = Vector(P_2, R_2)
v_2 = Vector(Q_2, A_6)
w_2 = Vector(P_2, C_6)
w'_2 = Vector(Q_2, R_2)
J = Intersect(q, r)
gB = Angle(Vector[B, C], Vector[B, A])
gA = Angle(Vector[A, B], Vector[A, C])
gC = Angle(Vector[C, A], Vector[C, B])
poly6, t, t_1, t_2, a_3, K, L = Polygon(C, A, 4)
poly7, b_3, c_3, d_3, e_3, M, N = Polygon(B, C, 4)
poly8, f_3, g_3, h_3, i_3, O, S = Polygon(A, B, 4)
T = Midpoint(L, A)
U = Midpoint(C, N)
V = Midpoint(A, O)
poly9, v_3, u_3, t_3 = Polygon(U, T, V)
j_3 = Segment(A, U)
k_3 = Segment(V, C)
l_3 = Segment(B, T)
A_7 = (-1.6014965986394545, -4.043292517006807)
B_7 = (4.340408163265296, -1.423628117913836)
C_7 = (2.280408163265301, 0.7163718820861655)
poly1_3, c_4, a_4, b_4 = Polygon(A_7, B_7, C_7)
poly2_3, d_4, e_4, f_4, g_4, D_3, E_3 = Polygon(A_7, C_7, 4)
poly3_3, h_4, i_4, j_4, k_4, F_3, G_3 = Polygon(B_7, A_7, 4)
poly4_3, l_4, m_3, n_3, p_7, H_3, I_3 = Polygon(C_7, B_7, 4)
R_3 = Midpoint(E_3, C_7)
Q_3 = Midpoint(B_7, I_3)
P_3 = Midpoint(F_3, B_7)
q_6 = Segment(C_7, P_3)
r_7 = Segment(Q_3, A_7)
s_3 = Segment(B_7, R_3)
poly5_3, p_6, r_6, q_7 = Polygon(R_3, Q_3, P_3)
u_4 = Vector(R_3, B_7)
u'_3 = Vector(P_3, Q_3)
su_3: u_4 * u'_3
v'_3 = Vector(P_3, R_3)
v_4 = Vector(Q_3, A_7)
w_3 = Vector(P_3, C_7)
w'_3 = Vector(Q_3, R_3)
A_8 = (-0.1286439909297049, -5.90246712018141)
B_8 = (5.813260770975056, -3.2828027210884443)
C_8 = (3.7532607709750563, -1.1428027210884517)
poly1_4, c_5, a_5, b_5 = Polygon(A_8, B_8, C_8)
poly2_4, d_5, e_5, f_5, g_5, D_4, E_4 = Polygon(A_8, C_8, 4)
poly3_4, h_5, i_5, j_5, k_5, F_4, G_4 = Polygon(B_8, A_8, 4)
poly4_4, l_5, m_4, n_4, p_9, H_4, I_4 = Polygon(C_8, B_8, 4)
R_4 = Midpoint(E_4, C_8)
Q_4 = Midpoint(B_8, I_4)
P_4 = Midpoint(F_4, B_8)
q_8 = Segment(C_8, P_4)
r_9 = Segment(Q_4, A_8)
s_4 = Segment(B_8, R_4)
poly5_4, p_8, r_8, q_9 = Polygon(R_4, Q_4, P_4)
u_5 = Vector(R_4, B_8)
u'_4 = Vector(P_4, Q_4)
su_4: u_5 * u'_4
v'_4 = Vector(P_4, R_4)
v_5 = Vector(Q_4, A_8)
w_4 = Vector(P_4, C_8)
w'_4 = Vector(Q_4, R_4)
A_9 = (14.792616780045352, -1.641995464852615)
B_9 = (20.7345215419501, 0.9776689342403562)
C_9 = (18.674521541950106, 3.117668934240358)
poly1_5, c_6, a_6, b_6 = Polygon(A_9, B_9, C_9)
poly2_5, d_6, e_6, f_6, g_6, D_5, E_5 = Polygon(A_9, C_9, 4)
poly3_5, h_6, i_6, j_6, k_6, F_5, G_5 = Polygon(B_9, A_9, 4)
poly4_5, l_6, m_5, n_5, p_{11}, H_5, I_5 = Polygon(C_9, B_9, 4)
R_5 = Midpoint(E_5, C_9)
Q_5 = Midpoint(B_9, I_5)
P_5 = Midpoint(F_5, B_9)
q_{10} = Segment(C_9, P_5)
r_{11} = Segment(Q_5, A_9)
s_5 = Segment(B_9, R_5)
poly5_5, p_{10}, r_{10}, q_{11} = Polygon(R_5, Q_5, P_5)
u_6 = Vector(R_5, B_9)
u'_5 = Vector(P_5, Q_5)
su_5: u_6 * u'_5
v'_5 = Vector(P_5, R_5)
v_6 = Vector(Q_5, A_9)
w_5 = Vector(P_5, C_9)
w'_5 = Vector(Q_5, R_5)
A_{10} = (7.308054421768707, -8.968897959183682)
B_{10} = (13.249959183673448, -6.349233560090706)
C_{10} = (11.189959183673464, -4.209233560090711)
poly1_6, c_7, a_7, b_7 = Polygon(A_{10}, B_{10}, C_{10})
poly2_6, d_7, e_7, f_7, g_7, D_6, E_6 = Polygon(A_{10}, C_{10}, 4)
poly3_6, h_7, i_7, j_7, k_7, F_6, G_6 = Polygon(B_{10}, A_{10}, 4)
poly4_6, l_7, m_6, n_6, p_{13}, H_6, I_6 = Polygon(C_{10}, B_{10}, 4)
R_6 = Midpoint(E_6, C_{10})
Q_6 = Midpoint(B_{10}, I_6)
P_6 = Midpoint(F_6, B_{10})
q_{12} = Segment(C_{10}, P_6)
r_{13} = Segment(Q_6, A_{10})
s_6 = Segment(B_{10}, R_6)
poly5_6, p_{12}, r_{12}, q_{13} = Polygon(R_6, Q_6, P_6)
u_7 = Vector(R_6, B_{10})
u'_6 = Vector(P_6, Q_6)
su_6: u_7 * u'_6
v'_6 = Vector(P_6, R_6)
v_7 = Vector(Q_6, A_{10})
w_6 = Vector(P_6, C_{10})
w'_6 = Vector(Q_6, R_6)
n_1: Q - P
z_RB: x(Vector[R, B]) + i * y(Vector[R, B])
α_1: P - Q
β_1: R - B
θ_1: α_1 / β_1
A_{11} = (-5.126684807256238, 3.948743764172338)
B_{11} = (1.7546757369614472, 2.35516553287982)
C_{11} = (-1.0220136054421847, 6.58056235827665)
poly1_7, c_8, a_8, b_8 = Polygon(A_{11}, B_{11}, C_{11})
poly2_7, d_8, e_8, f_8, g_8, D_7, E_7 = Polygon(A_{11}, C_{11}, 4)
poly3_7, h_8, i_8, j_8, k_8, F_7, G_7 = Polygon(B_{11}, A_{11}, 4)
poly4_1, l_1, m_1, n_8, p_3, H_1, I_1 = Polygon(C_{11}, B_{11}, 4)
R_7 = Midpoint(E_7, C_{11})
Q_1 = Midpoint(B_{11}, I_1)
P_7 = Midpoint(F_7, B_{11})
q_3 = Segment(C_{11}, P_7)
r_3 = Segment(Q_1, A_{11})
s_7 = Segment(B_{11}, R_7)
poly5_1, p_2, r_2, q_{14} = Polygon(R_7, Q_1, P_7)
u_8 = Vector(R_7, B_{11})
u'_1 = Vector(P_7, Q_1)
v'_7 = Vector(P_7, R_7)
v_1 = Vector(Q_1, A_{11})
w_7 = Vector(P_7, C_{11})
w'_1 = Vector(Q_1, R_7)
n_7: Q_1 - P_7
α_2: R - Q
β_2: P - C
θ_2: α_2 / β_2
α_3: R - P
β_3: Q - A
θ_3: α_3 / β_3
A_{12} = (-8.555292517006798, -9.693251700680285)
B_{12} = (-1.7705124716553522, -12.156054421768712)
C_{12} = (-5.561297052154202, -7.761641723356019)
poly1_8, c_9, a_9, b_9 = Polygon(A_{12}, B_{12}, C_{12})
poly2_8, d_9, e_9, f_9, g_9, D_8, E_8 = Polygon(A_{12}, C_{12}, 4)
poly3_8, h_9, i_9, j_9, k_9, F_8, G_8 = Polygon(B_{12}, A_{12}, 4)
poly4_7, l_8, m_7, n_{10}, p_{15}, H_7, I_7 = Polygon(C_{12}, B_{12}, 4)
R_8 = Midpoint(E_8, C_{12})
Q_7 = Midpoint(B_{12}, I_7)
P_8 = Midpoint(F_8, B_{12})
q_{15} = Segment(C_{12}, P_8)
r_{15} = Segment(Q_7, A_{12})
s_8 = Segment(B_{12}, R_8)
poly5_7, p_{14}, r_{14}, q_{16} = Polygon(R_8, Q_7, P_8)
u_9 = Vector(R_8, B_{12})
u'_7 = Vector(P_8, Q_7)
v'_8 = Vector(P_8, R_8)
v_8 = Vector(Q_7, A_{12})
w_8 = Vector(P_8, C_{12})
w'_7 = Vector(Q_7, R_8)
n_9: Q_7 - P_8
α_4: P_8 - Q_7
β_4: R_8 - B_{12}
θ_4: α_4 / β_4
α_6: R_8 - Q_7
β_5: P_8 - C_{12}
α_5: R_8 - P_8
β_6: Q_7 - A_{12}
texte2_2: "3)\;\;\frac{z_R-z_P}{z_Q-z_A}=\frac{\left( " + (LaTeX[R]) + " \right)-\left( " + (LaTeX[P]) + " \right)  }{\left(" + (LaTeX[Q]) + " \right)-\left(  " + (LaTeX[A]) + "\right)  }\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{" + (LaTeX[α_3]) + "}{" + (LaTeX[β_3]) + "}=" + (LaTeX[θ_3]) + ""
"\frac{z_P-z_Q}{z_R-z_B}=\frac{z_P-z_Q}{z_R-z_B}=\frac{\left(  " + (LaTeX[P_7]) + " \right)-\left( " + (LaTeX[Q_1]) + "  \right)  }{\left(" + (LaTeX[R_7]) + "   \right)-\left(  " + (LaTeX[B_{11}]) + " \right)  }\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{" + (LaTeX[α_2]) + "}{" + (LaTeX[β_2]) + "}=" + (LaTeX[θ_2]) + "": "2)\;\;\frac{z_R-z_Q}{z_P-z_C}=\frac{\left(  " + (LaTeX[R]) + " \right)-\left( " + (LaTeX[Q]) + " \right)  }{\left(" + (LaTeX[P]) + "  \right)-\left(  " + (LaTeX[C]) + "\right)  }\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{" + (LaTeX[α_2]) + "}{" + (LaTeX[β_2]) + "}=" + (LaTeX[θ_2]) + ""
texte2_3: "1)\;\;\frac{z_P-z_Q}{z_R-z_B}=\frac{\left(  " + (LaTeX[P]) + " \right)-\left( " + (LaTeX[Q]) + "  \right)  }{\left(" + (LaTeX[R]) + "   \right)-\left(  " + (LaTeX[B]) + " \right)  }\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{" + (LaTeX[α_1]) + "}{" + (LaTeX[β_1]) + "}=" + (LaTeX[θ_1]) + ""