\text{\bold{\textcolor{red}{X(897)\ }}\
Trilinears   \frac{1}{ 2a²-b²-c²} ,\frac{1}{ 2b²-c²-a²}, \frac{1}{ 2c²-a²-b²}\
1) X(897) = ISOGONAL CONJUGATE OF X(896)\
2) Let Ap_1Bp_1Cp_1 and Ap_3Bp_3Cp_3 be the 1st and 3rd Parry triangles. \
Let A' be the trilinear product Ap_1*Ap_3, \
and define B' and C' cyclically. \
\textcolor{red}{The lines AA', BB', CC' do not concur in X(897).}\

3) X(897) lies on these lines: \
1,662   10,190   19,162   37,100   \

4) X(897) = isotomic conjugate of X(14210)\
5) X(897) = anticomplement of X(16597)\
6) X(897) = trilinear pole of line X(1)X(661)\

\vspace{1.5cm}
Geogebra contributed by René Drucker, may 2025.\ 
}}}

Position(M,Séquence(TriangleCentre(A,B,C,n),n,1,5000))
X_{n}=TriangleCentre(A, B, C, n)
polar conjugate = 48-isoconjugate
X(30)X(511) is a line at infinity
\overrightarrow{X(782)}

X_{5}

35

33

20

55

69

$Euler\ line$

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

Lemoine axis

de Longchamp axis

trilinear

isogonal conjugate-896

X(14210)

isotomiconjugate-14210

X(16597)

Trilinear-pole-I-661

A1

B1

C1

A'1

B'1

C'1

Ap_1

Bp_1

Cp_1

Ap_3

Bp_3

Cp_3

A = (-23.689738260244486, 19.88432765853613)
B = (-53.97124685542226, -22.52298637012055)
C = (65.15758588063855, -21.803174691957047)
O = TriangleCenter(C, B, A, 3)
c' = Line(B, A)
ETC = 0 (slider from 0 to 50, step 1)
etc = TriangleCenter(C, B, A, ETC)
b' = Line(A, C)
a' = Line(C, B)
c = Segment(B, A)
b = Segment(A, C)
a = Segment(C, B)
c_O = Circle(C, B, A)
c_I = Incircle(C, B, A)
I = TriangleCenter(C, B, A, 1)
M_A = Midpoint(B, C)
M_B = Midpoint(A, C)
M_C = Midpoint(A, B)
H_A = ClosestPoint(a', A)
H_B = ClosestPoint(b', B)
H_C = ClosestPoint(c', C)
H = TriangleCenter(C, B, A, 4)
G = TriangleCenter(C, B, A, 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(C, H)
B_H = Midpoint(B, H)
h_B = Line(B, H_B)
N_9 = TriangleCenter(C, B, A, 5)
N_{ag} = TriangleCenter(C, B, A, 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)
bi_A = AngularBisector(B, A, C)
bi_B = AngularBisector(A, B, C)
bi_C = AngularBisector(A, C, B)
X_{35} = TriangleCenter(C, B, A, 35)
X_{33} = TriangleCenter(C, B, A, 33)
F_{eu} = TriangleCenter(C, B, A, 11)
L_e = TriangleCenter(C, B, A, 6)
M_{it} = TriangleCenter(C, B, A, 9)
G_e = TriangleCenter(C, B, A, 7)
X_{20} = TriangleCenter(C, B, A, 20)
L_{3,35} = Line(O, X_{35})
m_B = Line(B, M_B)
X_{55} = TriangleCenter(C, B, A, 55)
X_{69} = TriangleCenter(C, B, A, 69)
b_b = LineBisector(C, A)
X_{84} = TriangleCenter(C, B, A, 84)
X_{15} = TriangleCenter(C, B, A, 15)
X_{16} = TriangleCenter(C, B, A, 16)
X_{32} = TriangleCenter(C, B, A, 32)
X_{36} = TriangleCenter(C, B, A, 36)
X_{40} = TriangleCenter(C, B, A, 40)
X_{46} = TriangleCenter(C, B, A, 46)
L_{2,3} = Line(O, G)
X_{53} = TriangleCenter(C, B, A, 53)
X_{10} = TriangleCenter(C, B, A, 10)
X_{19} = TriangleCenter(C, B, A, 19)
X_{72} = TriangleCenter(C, B, A, 72)
X_{13} = TriangleCenter(C, B, A, 13)
X_{18} = TriangleCenter(C, B, A, 18)
X_{14} = TriangleCenter(C, B, A, 14)
X_{17} = TriangleCenter(C, B, A, 17)
L_{5,13} = Line(X_{13}, X_{18})
C_H = Midpoint(A, H)
X_{62} = TriangleCenter(C, B, A, 62)
X_{61} = TriangleCenter(C, B, A, 61)
X_{68} = TriangleCenter(C, B, A, 68)
X_{34} = TriangleCenter(C, B, A, 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(C, B, A, 21)
X_{31} = TriangleCenter(C, B, A, 31)
L_{8,21} = Line(X_{55}, N_{ag})
X_{24} = TriangleCenter(C, B, A, 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(C, B, A, 56)
X_{30} = TriangleCenter(C, B, A, 30)
n = 2 (slider from 0 to 6, 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(C, B, A, 48)
S: (2 * Area[C, B, A])
r_{1}: S / (c + b + a)
X_{336} = TriangleCenter(C, B, A, 336)
X_{75} = TriangleCenter(C, B, A, 75)
X_{73} = TriangleCenter(C, B, A, 73)
X_{290} = TriangleCenter(C, B, A, 290)
X_{255} = TriangleCenter(C, B, A, 255)
X_{293} = TriangleCenter(C, B, A, 293)
X_{1} = TriangleCenter(C, B, A, 240)
X_{240} = TriangleCenter(C, B, A, 240)
t'i_c = Line(Ti_B, Ti_A)
t'i_b = Line(Ti_C, Ti_A)
Bi_I = Intersect(bi_B, b')
S_B: (a^(2) + c^(2) - b^(2)) / 2
ω = BrocardAngle(C, B, A)
S_ω: (S * cot(ω))
s: (c + b + a) / 2
s_b: (a + c - b) / 2
R: (((c * b) * a)) / ((2 * S))
Δ = Area(C, B, A)
L_{1, 5} = Line(I, N_9)
X_{51} = TriangleCenter(C, B, A, 51)
X_{511} = Vector(G, X_{51})
L_{2,6} = Line(G, L_e)
X_{113} = TriangleCenter(C, B, A, 113)
X_{568} = TriangleCenter(C, B, A, 568)
be_A = OrthogonalLine(A, bi_A)
be_B = OrthogonalLine(B, bi_B)
be_C = OrthogonalLine(C, bi_C)
Ie_B = Intersect(be_A, be_C)
Ie_A = Intersect(be_C, be_B)
Ie_C = Intersect(be_B, be_A)
X_{709} = TriangleCenter(C, B, A, 709)
b_a = LineBisector(B, C)
m_A = Line(A, M_A)
m_C = Line(C, M_C)
h_C = Line(C, H_C)
h_A = Line(A, H_A)
b_c = LineBisector(B, A)
Te_A = ClosestPoint(a, Ie_A)
Te_B = ClosestPoint(b, Ie_B)
Te_C = ClosestPoint(c, Ie_C)
ce_A = Circle(Ie_A, Te_A)
ce_B = Circle(Ie_B, Te_B)
ce_C = Circle(Ie_C, Te_C)
α = Angle(B, A, C)
γ = Angle(A, C, B)
β = Angle(A, B, C)
s_a: (b - a + c) / 2
s_c: ((-c) + b + a) / 2
Ai_I = Intersect(bi_A, a')
Ci_I = Intersect(bi_C, c')
Pos = IndexOf(I, Sequence[TriangleCenter[A, B, C, n], n, 1, 5000])
P = (-83.48352566012606, 57.376109906028155)
U = (-78.4448439129815, 30.023266135815074)
Tril_P = CoordTrilinABCP(A, B, C, P)
Tril_U = CoordTrilinABCP(A, B, C, U)
p_1 = Element(Tril_P, 1)
p_2 = Element(Tril_P, 2)
p_3 = Element(Tril_P, 3)
u_1 = Element(Tril_U, 1)
u_2 = Element(Tril_U, 2)
u_3 = Element(Tril_U, 3)
X_{835} = TriangleCenter(A, B, C, 835)
X_{58} = TriangleCenter(A, B, C, 58)
X_{325} = TriangleCenter(A, B, C, 325)
X_{684} = TriangleCenter(A, B, C, 684)
X_{187} = TriangleCenter(A, B, C, 187)
X_{237} = TriangleCenter(A, B, C, 237)
L_{Lemoine} = Line(X_{237}, X_{187})
L_{Longchamp} = Line(X_{684}, X_{325})
S_A: (b^(2) - a^(2) + c^(2)) / 2
S_C: ((-c^(2)) + b^(2) + a^(2)) / 2
obtu = If(Max[{α, β, γ}] > pi / 2, 1, 0)
X_{74} = TriangleCenter(A, B, C, 74)
X_{99} = TriangleCenter(A, B, C, 99)
X_{892} = TriangleCenter(A, B, C, 892)
X_{896} = TriangleCenter(A, B, C, 896)
X_{12} = TriangleCenter(A, B, C, 12)
D = Trilinear(A, B, C, 1 / ((2 * a^(2)) - b^(2) - c^(2)), 1 / ((2 * b^(2)) - c^(2) - a^(2)), 1 / ((2 * c^(2)) - a^(2) - b^(2)))
X_{897} = TriangleCenter(A, B, C, 897)
E, f, g, h, i, j, k, l, m, p = IsogonalConjugate(A, B, C, X_{896})
X_{662} = TriangleCenter(A, B, C, 662)
X_{190} = TriangleCenter(A, B, C, 190)
X_{162} = TriangleCenter(A, B, C, 162)
X_{37} = TriangleCenter(A, B, C, 37)
X_{100} = TriangleCenter(A, B, C, 100)
q = Line(X_{19}, X_{162})
r = Line(X_{662}, I)
t = Line(X_{37}, X_{100})
d = Line(X_{10}, X_{190})
F = X14210(A, B, C)
e, A_{14210}, f_1, A'_{14210}, B'_{14210}, g_1, h_1, B_{14210}, i_1, C_{14210}, j_1, C'_{14210}, T = IsotomicConjugate(A, B, C, F)
J = X16597(A, B, C)
X_{661} = TriangleCenter(A, B, C, 661)
K, K_{2}, K_{3}, K_{4}, K_{5}, K_{6}, K_{7} = trilinearpolePQ(A, B, C, I, X_{661})
k_1 = Line(I, X_{661})
L, M, N = ParryTriangle11(A, B, C)
Q, V, W = ParryTriangle31(A, B, C)
A' = Trilinearproduct(A, B, C, (A * p_1), (A * p_3))
B' = Trilinearproduct(A, B, C, (B * p_1), (B * p_3))
C' = Trilinearproduct(A, B, C, (C * p_1), (C * p_3))
l_1 = Line(B, B')
m_1 = Line(A, A')
n_1 = Line(C, C')
Z = Intersect(n_1, l_1)
pos = IndexOf(Z, Sequence[TriangleCenter[A, B, C, n], n, 1, 5000])

Tool: AntiCevianTriangle
sommetsX3, point
Input: A, B, C, P
Output: A', B', C'

B'_P

A'_P

C'_P

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
f = Line(A, P)
g = Line(B, P)
h = Line(C, P)
C_P = Intersect(h, c)
B_P = Intersect(b, g)
A_P = Intersect(f, a)
b_P = Line(C_P, A_P)
a_P = Line(C_P, B_P)
c_P = Line(B_P, A_P)
B_0 = Intersect(b, b_P)
A_0 = Intersect(a, a_P)
C_0 = Intersect(c, c_P)
l = Line(A_0, A)
m = Line(C_0, C)
n = Line(B_0, B)
B' = Intersect(m, l)
A' = Intersect(m, n)
C' = Intersect(n, l)

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

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

Tool: CevaConjugate (command CevaConjugate1)
sommetsX3, pointsX2
Input: A, B, C, P, Q
Output: X, A_P, B_P, C_P, A'_Q, C'_Q, B'_Q, f, g, h

Ap

Bp

Cp

A'q

B'q

C'q

P-Ceva Conjugate-Q

a' = Line(B, C)
b' = Line(C, A)
c' = Line(A, B)
u = Element(Tril_U, 1)
v = Element(Tril_U, 2)
w = Element(Tril_U, 3)
A_P = Intersect(Line[A, P], a')
B_P = Intersect(Line[B, P], b')
C_P = Intersect(Line[C, P], c')
A'_Q = Trilinear(A, B, C, (-u), v, w)
B'_Q = Trilinear(A, B, C, u, (-v), w)
C'_Q = Trilinear(A, B, C, u, v, (-w))
f = Line(B_P, B'_Q)
g = Line(A'_Q, A_P)
h = Line(C_P, C'_Q)
X = Intersect(f, g)

Tool: cevapointPU
sommetsX3, pointsX2
Input: A, B, C, P, U
Output: A_P, B_P, Pc, Au, Bu, Cu, i_3, g_3, h_3, Apu, Bpu, Cpu, X

euler

Ua

Ub

Uc

X=Cevapoint-P-U

Pa

Pb

A_{pu}

B_{pu}

C_{pu}

p = Element(Tril_P, 1)
Au = (20.143315588586628, -3.3952641427394212)
Bu = (22.959311224829445, 4.8819849768790196)
Cu = (11.169588797719653, 10.11848261240921)
q = Element(Tril_P, 2)
r = Element(Tril_P, 3)
u = Element(Tril_U, 1)
v = Element(Tril_U, 2)
w = Element(Tril_U, 3)
X = Trilinear(A, B, C, (((p * v) + (q * u)) * ((p * w) + (r * u))), (((q * w) + (r * v)) * ((q * u) + (p * v))), (((r * u) + (p * w)) * ((r * v) + (q * w))))
A_P = (8.196777171908499, -2.7227811328021407)
B_P = (20.674755213196942, 8.980664458307974)
Pc = (7.15129740329277, 5.780387631338337)
g_3 = Line(Au, Pc)
h_3 = Line(A_P, Bu)
Apu = Intersect(h_3, g_3)
i_3 = Line(B_P, Cu)
Bpu = Intersect(i_3, h_3)
Cpu = Intersect(i_3, g_3)

Tool: Ceviantriangle
sommetsx3-Point
Input: A, B, C, P
Output: A_P, B_P, C_P, k, i, j, g, h, f

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
f = Line(A, P)
g = Line(B, P)
h = Line(C, P)
C_P = Intersect(h, c)
B_P = Intersect(b, g)
A_P = Intersect(f, a)
i = Segment(A_P, B_P)
j = Segment(B_P, C_P)
k = Segment(C_P, A_P)

Tool: CrossPoint
sommetsX3, pointsX2
Input: A, B, C, P, U
Output: X, A_P, B_P, C_P, g, h, f, C_U, B_U, A_U, j_1, l_1, k_1, A_i, Bi, Ci, n_1, m_1, p_1

euler

p = Element(Tril_P, 1)
U: (22, 24)
q = Element(Tril_P, 2)
u = Element(Tril_U, 1)
v = Element(Tril_U, 2)
w = Element(Tril_U, 3)
r = Element(Tril_P, 3)
X = Trilinear(A, B, C, (p * (u * ((r * v) + (q * w)))), (q * (v * ((p * w) + (r * u)))), (r * (w * ((q * u) + (p * v)))))
A_P, B_P, C_P, f, g, h, i, j, k = Ceviantriangle(A, B, C, P)
A_U, B_U, C_U, l, m, t, d, e, f_1 = Ceviantriangle(A, B, C, U)
g_1 = Line(B_P, C_P)
h_1 = Line(A_P, C_P)
i_1 = Line(B_P, A_P)
j_1 = Line(A, A_U)
k_1 = Line(B, B_U)
l_1 = Line(C, C_U)
Bi = Intersect(k_1, h_1)
Ci = Intersect(l_1, i_1)
A_i = Intersect(j_1, g_1)
m_1 = Line(A_P, A_i)
n_1 = Line(Ci, C_P)
p_1 = Line(Bi, B_P)

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: CrossConjugate
sommetsX3, pointsX2
Input: A, B, C, P, U
Output: X_{pu}, A', B', C', A'', C'', B'', l_2, m_2, p_2

A_U

B_U

C_U

A''_P

B''_P

C''_P

Xpu

A' = (13.808309086157802, -2.0078820801716075)
B' = (33.0402360663662, 9.424934122292303)
C' = (11.453106409938568, 7.638717287031762)
p_1 = Line(C', B')
q_1 = Line(C', A')
r_1 = Line(A', B')
s_1 = Line(P, A')
a_1 = Line(P, B')
k_2 = Line(P, C')
A'' = Intersect(p_1, s_1)
B'' = Intersect(a_1, q_1)
C'' = Intersect(k_2, r_1)
l_2 = Line(A, A'')
m_2 = Line(B, B'')
p_2 = Line(C, C'')
X_{pu} = Intersect(l_2, m_2)

Tool: CrossSumPQ
sommetsx3-Point-Point
Input: A, B, C, P, Q
Output: W

l1 = CoordTrilinABCP(A, B, C, P)
l2 = CoordTrilinABCP(A, B, C, Q)
w1: (Element[l1, 2] * Element[l2, 3]) + (Element[l1, 3] * Element[l2, 2])
w2: (Element[l1, 1] * Element[l2, 3]) + (Element[l1, 3] * Element[l2, 1])
w3: (Element[l1, 2] * Element[l2, 1]) + (Element[l1, 1] * Element[l2, 2])
W = Trilinear(A, B, C, w1, w2, w3)

Tool: CyclocevianConjugate
sommetsx3-Point
Input: A, B, C, P
Output: P', P_C, P_B, P_A, r, q, s, e, P'_B, P'_A, P'_C, g_1, t, f_1

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
P_A = (10.819051225863577, -4.010558248015659)
P_B = (10.777399809699789, -1.0545619083735878)
P_C = (7.129593114981156, -1.1396577641459007)
e = Circle(P_B, P_C, P_A)
O_P = Center(e)
P'_A = Mirror(P_A, OrthogonalLine[O_P, a])
P'_B = Mirror(P_B, OrthogonalLine[O_P, b])
P'_C = Mirror(P_C, OrthogonalLine[O_P, c])
t = Line(B, P'_B)
f_1 = Line(A, P'_A)
g_1 = Line(C, P'_C)
P' = Intersect(t, g_1)

Tool: cyclotomicConjugate
sommetsX3, point
Input: P, A, B, C
Output: Q

cyclotomic conjugate

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
P_A = ClosestPoint(a, P)
P_B = ClosestPoint(b, P)
P_C = ClosestPoint(c, P)
d = Circle(P_B, P_C, P_A)
O_P = Center(d)
P'_A = Mirror(P_A, OrthogonalLine[O_P, a])
P'_B = Mirror(P_B, OrthogonalLine[O_P, b])
f = OrthogonalLine(P'_A, a)
g = OrthogonalLine(P'_B, b)
Q = Intersect(f, g)

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')

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: triangleCevian
sommetsx3-Point
Input: A, B, C, P
Output: A_P, B_P, C_P, k, i, j, g, h, f

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
f = Line(A, P)
g = Line(B, P)
h = Line(C, P)
C_P = Intersect(h, c)
B_P = Intersect(b, g)
A_P = Intersect(f, a)
i = Segment(A_P, B_P)
j = Segment(B_P, C_P)
k = Segment(C_P, A_P)

Tool: X4892
sommetsX3
Input: A, B, C
Output: X_{4892}

S_p

X(4892)

G = TriangleCenter(C, B, A, 2)
X_{10} = TriangleCenter(C, B, A, 10)
X_{896} = TriangleCenter(A, B, C, 896)
X_{12} = TriangleCenter(A, B, C, 12)
f = Line(X_{896}, G)
g = Line(X_{12}, X_{10})
X_{4892} = Intersect(g, f)

Tool: IsotomicConjugate (command IsotomicConjugate1)
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: X14210
sommetX3
Input: A, B, C
Output: X_{14210}

X(14210)

X_{897} = TriangleCenter(A, B, C, 897)
f, D, g, E, F, h, i, J, j, K, k, L, X_{14210} = IsotomicConjugate(A, B, C, X_{897})

Tool: X16597
sommetsX3
Input: A, B, C
Output: X_{16597}

X(16597)

F_{eu} = TriangleCenter(C, B, A, 11)
X_{126} = TriangleCenter(A, B, C, 126)
X_{1086} = TriangleCenter(A, B, C, 1086)
X_{1125} = TriangleCenter(A, B, C, 1125)
g = Line(X_{1086}, X_{1125})
h = Line(F_{eu}, X_{126})
X_{16597} = Intersect(g, h)

Tool: ConjuguéHarmonique
extrêmité,extrêmité,Point
Input: C, B, P
Output: P'

S = Rotate(C, 60°, B)
f = Line(S, C)
h = Line(C, B)
i = Line(S, P)
j = Line(B, f)
M = Intersect(i, j)
M' = Mirror(M, B)
k = Line(S, M')
P' = Intersect(k, h)

Tool: trilinearpolePQ
sommetsX3, pointsX2
Input: A, B, C, P, Q
Output: K, A', B', C', A'', B'', C''

A1

C1

B1

A2

B2

C2

Trilinear-pole-P-Q

a = Line(B, C)
l_3 = Line(P, Q)
A' = Intersect(a, l_3)
b = Line(C, A)
c = Line(A, B)
C' = Intersect(c, l_3)
B' = Intersect(l_3, b)
A'' = ConjuguéHarmonique(B, C, A')
B'' = ConjuguéHarmonique(A, C, B')
C'' = ConjuguéHarmonique(B, A, C')
e_3 = Line(B, B'')
f_4 = Line(C, C'')
K = Intersect(f_4, e_3)

Tool: Cercle2Parry
sommetsx3
Input: A, B, C
Output: d, X_{351}, X_{16}, X_{23}, X_{111}, X_{110}

G = TriangleCenter(A, B, C, 2)
X_{110} = TriangleCenter(A, B, C, 110)
X_{111} = TriangleCenter(A, B, C, 111)
X_{15} = TriangleCenter(A, B, C, 15)
X_{16} = TriangleCenter(A, B, C, 16)
X_{23} = TriangleCenter(A, B, C, 23)
d = Circle(G, X_{15}, X_{16})
X_{351} = TriangleCenter(A, B, C, 351)

Tool: ParryTriangle1
sommetsx3
Input: A, B, C
Output: s_1, t_3, a_3, B1, A1, C1

X_{111} = TriangleCenter(A, B, C, 111)
p_1, O_P, F, J, K, M = Cercle2Parry(A, B, C)
A1 = Mirror(X_{111}, OrthogonalLine[O_P, Line[A, X_{111}]])
B1 = Mirror(X_{111}, OrthogonalLine[O_P, Line[B, X_{111}]])
C1 = Mirror(X_{111}, OrthogonalLine[O_P, Line[C, X_{111}]])
s_1 = Segment(A1, B1)
t_3 = Segment(B1, C1)
a_3 = Segment(C1, A1)

Tool: Cercle2Parry (command Cercle2Parry1)
sommetsx3
Input: A, B, C
Output: d, X_{351}, X_{16}, X_{23}, X_{111}, X_{110}

G = TriangleCenter(A, B, C, 2)
X_{110} = TriangleCenter(A, B, C, 110)
X_{111} = TriangleCenter(A, B, C, 111)
X_{15} = TriangleCenter(A, B, C, 15)
X_{16} = TriangleCenter(A, B, C, 16)
X_{23} = TriangleCenter(A, B, C, 23)
d = Circle(G, X_{15}, X_{16})
X_{351} = TriangleCenter(A, B, C, 351)

Tool: ParryTriangle3
sommetsx3
Input: A, B, C
Output: A3, a_3, s_1, C3, t_3, B3

G = TriangleCenter(A, B, C, 2)
p_1, O_P, F, J, K, M = Cercle2Parry(A, B, C)
A3 = Mirror(G, OrthogonalLine[O_P, Line[A, G]])
B3 = Mirror(G, OrthogonalLine[O_P, Line[B, G]])
C3 = Mirror(G, OrthogonalLine[O_P, Line[C, G]])
s_1 = Segment(A3, B3)
t_3 = Segment(B3, C3)
a_3 = Segment(C3, A3)

Tool: Cercle2Parry (command Cercle2Parry2)
sommetsx3
Input: A, B, C
Output: d, X_{351}, X_{16}, X_{23}, X_{111}, X_{110}

G = TriangleCenter(A, B, C, 2)
X_{110} = TriangleCenter(A, B, C, 110)
X_{111} = TriangleCenter(A, B, C, 111)
X_{15} = TriangleCenter(A, B, C, 15)
X_{16} = TriangleCenter(A, B, C, 16)
X_{23} = TriangleCenter(A, B, C, 23)
d = Circle(G, X_{15}, X_{16})
X_{351} = TriangleCenter(A, B, C, 351)

Tool: ParryTriangle1 (command ParryTriangle11)
sommetsX3
Input: A, B, C
Output: A1, B1, C1

Ap_1

Bp_1

Cp_1

X_{111} = TriangleCenter(A, B, C, 111)
p_1, O_P, F, J, K, M = Cercle2Parry(A, B, C)
A1 = Mirror(X_{111}, OrthogonalLine[O_P, Line[A, X_{111}]])
B1 = Mirror(X_{111}, OrthogonalLine[O_P, Line[B, X_{111}]])
C1 = Mirror(X_{111}, OrthogonalLine[O_P, Line[C, X_{111}]])

Tool: Cercle2Parry (command Cercle2Parry3)
sommetsx3
Input: A, B, C
Output: d, X_{351}, X_{16}, X_{23}, X_{111}, X_{110}

G = TriangleCenter(A, B, C, 2)
X_{110} = TriangleCenter(A, B, C, 110)
X_{111} = TriangleCenter(A, B, C, 111)
X_{15} = TriangleCenter(A, B, C, 15)
X_{16} = TriangleCenter(A, B, C, 16)
X_{23} = TriangleCenter(A, B, C, 23)
d = Circle(G, X_{15}, X_{16})
X_{351} = TriangleCenter(A, B, C, 351)

Tool: ParryTriangle3 (command ParryTriangle31)
sommetsX3
Input: A, B, C
Output: A3, B3, C3

Ap_3

Bp_3

Cp_3

G = TriangleCenter(A, B, C, 2)
p_1, O_P, F, J, K, M = Cercle2Parry(A, B, C)
A3 = Mirror(G, OrthogonalLine[O_P, Line[A, G]])
B3 = Mirror(G, OrthogonalLine[O_P, Line[B, G]])
C3 = Mirror(G, OrthogonalLine[O_P, Line[C, G]])

Tool: listTrilPoint
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: Trilinearproduct
sommetsX3
Input: A, B, C, P, Q
Output: W

P-trilinear product-Q

Plist = listTrilPoint(A, B, C, P)
Qlist = listTrilPoint(A, B, C, Q)
PQlist1: (Element[Plist, 1] * Element[Qlist, 1])
PQlist2: (Element[Plist, 2] * Element[Qlist, 2])
PQlist3: (Element[Plist, 3] * Element[Qlist, 3])
W = Trilinear(A, B, C, PQlist1, PQlist2, PQlist3)