Le \;point \;de\; Vecten\;\;\;\;\left(  Produit\;scalaire \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

A = (2.4790294784580515, -1.2424580498866242)
B = (8.684326530612246, -0.4939591836734718)
C = (4.555510204081634, 1.848117913832199)
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)
poly4_1, l_1, m_1, n_1, p_3, H_1, I_1 = Polygon(C_5, B_5, 4)
R_1 = Midpoint(E_1, C_5)
Q_1 = Midpoint(B_5, I_1)
P_1 = Midpoint(F_1, B_5)
q_2 = Segment(C_5, P_1)
r_3 = Segment(Q_1, A_5)
s_1 = Segment(B_5, R_1)
poly5_1, p_2, r_2, q_3 = Polygon(R_1, Q_1, P_1)
u_1 = Vector(R_1, B_5)
u'_1 = Vector(P_1, Q_1)
su_1: u_1 * u'_1
v'_1 = Vector(P_1, R_1)
v_1 = Vector(Q_1, A_5)
w_1 = Vector(P_1, C_5)
w'_1 = Vector(Q_1, R_1)
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)
texte2: "\overrightarrow{QA}" + (LaTeX[v]) + "\;et\;\overrightarrow{PR}" + (LaTeX[v']) + "\;alors\;\overrightarrow{QA}.\overrightarrow{PR}=" + (LaTeX[sv]) + "\\
alors\;ces\;deux\;vecteurs\;sont\;orthogonaux\\
donc\;(QA)\; est\; la\; hauteur\; issue\; de\; Q\; du\; triangle \;PQR"
texte2_2: "\overrightarrow{RB}" + (LaTeX[u_2]) + "\;et\;\overrightarrow{PQ}" + (LaTeX[u'_2]) + "\;alors\;\overrightarrow{RB}.\overrightarrow{PQ}=" + (LaTeX[su_2]) + "\\
alors\;ces\;deux\;vecteurs\;sont\;orthogonaux\\
donc\;(RB)\; est\; la\; hauteur\; issue\; de\; R\; du\; triangle \;PQR"
texte2_1: "\overrightarrow{PC}" + (LaTeX[w]) + "\;et\;\overrightarrow{QR}" + (LaTeX[w']) + "\;alors\;\overrightarrow{PC}.\overrightarrow{QR}=" + (LaTeX[sw]) + "\\
alors\;ces\;deux\;vecteurs\;sont\;orthogonaux\\
donc\;(PC)\; est\; la\; hauteur\; issue\; de\; P\; du\; triangle \;PQR"
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)