\text{\bold{\textcolor{red}{X(601)\ }}\
Trilinears  sin2A + cos A : sin2B + cos B : sin2C + cos C\
1) X(601) = INTERSECTION OF LINES 3, 31 AND 55, 255\
X(601) lies on these lines: \
1,104   \
3,31   \
4,171   \


\vspace{1.5cm}
Geogebra contributed by René Drucker, june 2024.\ 
}}

Position(M,Séquence(TriangleCentre(A,B,C,n),n,1,5000))
X_{n}=TriangleCentre(A, B, C, n)
polar conjugate = 48-isoconjugate

X_{5}

euler

35

33

20

55

69

Euler

S_p

L(5,13)

Brocard axis

Van Aubel line

Fermat line

Napoléon axis

Nagel

S_{ch}

(O-I) line

Soddy line

I, N_9 line

X(511)

(G, Le) line

P = (-22.081302181596783, 20.223681944868257)
C = (77.3224778879071, -2.774741606785337)
B = (-7.356731332596738, -3.4843800422107085)
A = (14.902026630733376, 46.535656610851326)
Tril_P = CoordTrilinABCP(A, B, C, P)
O = TriangleCenter(A, B, C, 3)
a' = Line(B, C)
ETC = 0 (slider from 0 to 50, step 1)
etc = TriangleCenter(A, B, C, ETC)
b' = Line(C, A)
c' = Line(A, B)
a = Segment(B, C)
b = Segment(C, A)
c = Segment(A, B)
c_O = Circle(A, B, C)
c_I = Incircle(A, B, C)
I = TriangleCenter(A, B, C, 1)
M_A = Midpoint(B, C)
M_B = Midpoint(C, A)
M_C = Midpoint(A, B)
H_A = ClosestPoint(a', A)
H_B = ClosestPoint(b', B)
H_C = ClosestPoint(c', C)
H = TriangleCenter(A, B, C, 4)
G = TriangleCenter(A, B, C, 2)
Ti_A = ClosestPoint(a', I)
Ti_B = ClosestPoint(b', I)
Ti_C = ClosestPoint(c', I)
c_9 = Circle(M_B, M_A, M_C)
A_H = Midpoint(A, H)
B_H = Midpoint(B, H)
h_B = Line(B, H_B)
h_C = Line(C, H_C)
N_9 = TriangleCenter(A, B, C, 5)
p_1 = Element(Tril_P, 1)
N_{ag} = TriangleCenter(A, B, C, 8)
t_A = Tangent(A, c_O)
t_B = Tangent(B, c_O)
t_C = Tangent(C, c_O)
C_e = Intersect(t_B, t_A)
A_e = Intersect(t_B, t_C)
B_e = Intersect(t_A, t_C)
t'i_a = Line(Ti_B, Ti_C)
k_c = Line(Ti_A, Ti_B)
h'_A = Line(H_B, H_C)
h'_b = Line(H_C, H_A)
h'_c = Line(H_A, H_B)
lieu_{etc} = Locus(etc, ETC)
b_A = AngularBisector(B, A, C)
b_B = AngularBisector(C, B, A)
b_C = AngularBisector(A, C, B)
X_{35} = TriangleCenter(A, B, C, 35)
X_{33} = TriangleCenter(A, B, C, 33)
F_{eu} = TriangleCenter(A, B, C, 11)
L_e = TriangleCenter(A, B, C, 6)
M_{it} = TriangleCenter(A, B, C, 9)
G_e = TriangleCenter(A, B, C, 7)
X_{20} = TriangleCenter(A, B, C, 20)
L_{3,35} = Line(O, X_{35})
m_A = Line(A, M_A)
m_B = Line(B, M_B)
m_C = Line(C, M_C)
X_{55} = TriangleCenter(A, B, C, 55)
X_{69} = TriangleCenter(A, B, C, 69)
b_a = LineBisector(B, C)
b_b = LineBisector(A, C)
b_c = LineBisector(B, A)
X_{84} = TriangleCenter(A, B, C, 84)
X_{15} = TriangleCenter(A, B, C, 15)
X_{16} = TriangleCenter(A, B, C, 16)
X_{32} = TriangleCenter(A, B, C, 32)
X_{36} = TriangleCenter(A, B, C, 36)
X_{40} = TriangleCenter(A, B, C, 40)
X_{46} = TriangleCenter(A, B, C, 46)
L_{2,3} = Line(O, G)
h_A = Line(A, H_A)
X_{53} = TriangleCenter(A, B, C, 53)
X_{10} = TriangleCenter(A, B, C, 10)
X_{19} = TriangleCenter(A, B, C, 19)
X_{72} = TriangleCenter(A, B, C, 72)
X_{13} = TriangleCenter(A, B, C, 13)
X_{18} = TriangleCenter(A, B, C, 18)
X_{14} = TriangleCenter(A, B, C, 14)
X_{17} = TriangleCenter(A, B, C, 17)
L_{5,13} = Line(X_{13}, X_{18})
C_H = Midpoint(C, H)
X_{62} = TriangleCenter(A, B, C, 62)
X_{61} = TriangleCenter(A, B, C, 61)
X_{68} = TriangleCenter(A, B, C, 68)
X_{34} = TriangleCenter(A, B, C, 34)
L_{1,6} = Line(I, M_{it})
L_{3,6} = Line(O, X_{15})
L_{4,6} = Line(H, X_{53})
L_{6,13} = Line(X_{13}, X_{14})
L_{6,17} = Line(L_e, X_{18})
L_{6,19} = Line(L_e, X_{19})
L_{1,2} = Line(N_{ag}, I)
L_{4,8} = Line(H, N_{ag})
L_{8,20} = Line(N_{ag}, X_{20})
X_{21} = TriangleCenter(A, B, C, 21)
X_{31} = TriangleCenter(A, B, C, 31)
L_{8,21} = Line(X_{55}, N_{ag})
X_{24} = TriangleCenter(A, B, C, 24)
L_{1,3} = Line(I, O)
L_{1,4} = Line(I, H)
L_{1,7} = Line(I, G_e)
L_{2,7} = Line(G, G_e)
L_{2,11} = Line(G, M_{it})
L_{3,9} = Line(O, M_{it})
L_{4,9} = Line(H, M_{it})
L_{5,6} = Line(N_9, L_e)
L_{7,8} = Line(G_e, N_{ag})
L_{9,21} = Line(M_{it}, X_{21})
L_{24,33} = Line(X_{24}, X_{33})
X_{56} = TriangleCenter(A, B, C, 56)
X_{30} = TriangleCenter(A, B, C, 30)
n = 0 (slider from 0 to 2, step 1)
L_{2,17} = Line(G, X_{62})
L_{3,13} = Line(O, X_{13})
L_{4,15} = Line(H, X_{15})
X_{48} = TriangleCenter(A, B, C, 48)
U = (-2.8461386863709914, 44.00048126535551)
Tril_U = CoordTrilinABCP(A, B, C, U)
p_2 = Element(Tril_P, 2)
S: (2 * Area[A, B, C])
r: S / (a + b + c)
u_1 = Element(Tril_U, 1)
u_2 = Element(Tril_U, 2)
u_3 = Element(Tril_U, 3)
X_{336} = TriangleCenter(A, B, C, 336)
X_{75} = TriangleCenter(A, B, C, 75)
X_{73} = TriangleCenter(A, B, C, 73)
X_{290} = TriangleCenter(A, B, C, 290)
X_{255} = TriangleCenter(A, B, C, 255)
X_{293} = TriangleCenter(A, B, C, 293)
X_{1} = TriangleCenter(A, B, C, 240)
X_{240} = TriangleCenter(A, B, C, 240)
t'i_c = Line(Ti_B, Ti_A)
t'i_b = Line(Ti_C, Ti_A)
Ai_I = Intersect(b_A, a')
Ci_I = Intersect(b_C, c')
Bi_I = Intersect(b_B, b')
S_A: (b^(2) + c^(2) - a^(2)) / 2
S_B: (c^(2) + a^(2) - b^(2)) / 2
S_C: (a^(2) + b^(2) - c^(2)) / 2
ω = Angle2Brocard(A, B, C)
S_ω: (S * cot(ω))
s: (a + b + c) / 2
s_a: (b + c - a) / 2
s_b: (c + a - b) / 2
s_c: (a + b - c) / 2
R: (((a * b) * c)) / ((2 * S))
Δ = Area(A, B, C)
p_3 = Element(Tril_P, 3)
α = Angle(B, A, C)
β = Angle(C, B, A)
γ = Angle(A, C, B)
L_{1, 5} = Line(I, N_9)
X_{51} = TriangleCenter(A, B, C, 51)
X_{511} = Vector(G, X_{51})
L_{2,6} = Line(G, L_e)
X_{113} = TriangleCenter(A, B, C, 113)
X_{568} = TriangleCenter(A, B, C, 568)
X_{601} = TriangleCenter(A, B, C, 601)
X_{104} = TriangleCenter(A, B, C, 104)
X_{171} = TriangleCenter(A, B, C, 171)
f = Line(X_{31}, O)
g = Line(X_{55}, X_{255})
h = Line(H, X_{171})
i = Line(X_{104}, I)

Tool: CoordTrilinABCP
sommetsx3,Point
Input: A, B, C, P
Output: l1

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
H_A = ClosestPoint(a, A)
H_B = ClosestPoint(b, B)
H_C = ClosestPoint(c, C)
P_A = ClosestPoint(a, P)
P_B = ClosestPoint(b, P)
P_C = ClosestPoint(c, P)
h_a = Vector(H_A, A)
h_b = Vector(H_B, B)
h_c = Vector(H_C, C)
p_b = Vector(P_B, P)
p_c = Vector(P_C, P)
p_a = Vector(P_A, P)
a_P: (p_a * h_a / Distance[A, H_A])
c_P: (p_c * h_c / Distance[C, H_C])
b_P: (p_b * h_b / Distance[B, H_B])
l1: {a_P, b_P, c_P}

Tool: IsotomicConjugate
sommetsx3-Point
Input: A, B, C, M
Output: e, A_M, i, A'_M, B'_M, j, f, B_M, g, C_M, h, C'_M, M'

c = Line(A, B)
a = Line(B, C)
b = Line(C, A)
f = Line(B, M)
e = Line(A, M)
g = Line(C, M)
A_M = Intersect(a, e)
B_M = Intersect(b, f)
C_M = Intersect(c, g)
A'_M = Mirror(A_M, Midpoint[B, C])
B'_M = Mirror(B_M, Midpoint[A, C])
C'_M = Mirror(C_M, Midpoint[A, B])
h = Line(C, C'_M)
i = Line(A, A'_M)
M' = Intersect(h, i)
j = Line(B, B'_M)

Tool: CercleDiamètre
Pointsx2
Input: B, A
Output: c_4, E

E = Midpoint(A, B)
c_4 = Circle(E, B)

Tool: Angle2Brocard
sommetsx3
Input: A, B, C
Output: ω

a = Line(B, C)
c = Line(A, B)
q_1 = OrthogonalLine(B, a)
a_1 = LineBisector(A, B)
O_{AB} = Intersect(q_1, a_1)
k_2 = LineBisector(A, C)
l_2 = OrthogonalLine(A, c)
O_{AC} = Intersect(k_2, l_2)
Br_1 = Mirror(A, Line[O_{AC}, O_{AB}])
ω = Angle(B, A, Br_1)

Tool: IsogonalConjugate
sommetsx3-Point
Input: A, B, C, P
Output: P', i', k', j', g, f, h, k, j, i

f = AngularBisector(A, B, C)
g = AngularBisector(B, C, A)
h = AngularBisector(C, A, B)
i = Line(A, P)
i' = Mirror(i, h)
j = Line(B, P)
j' = Mirror(j, f)
k = Line(P, C)
k' = Mirror(k, g)
P' = Intersect(j', i')