\text{\bold{\textcolor{red}{X(265)\ }}\
1) X(265) = REFLECTION OF X(3) IN X(125)\

2) Let A't be the reflection in line BC of the A-vertex of \
the tangential triangle, and define B't and C't cyclically. \
The circumcircles of A'tBC, B'tCA, and C'tAB concur in X(265). \
3) Let A'_5 be the reflection in line BC of the A-vertex = A_5 of \
the anticevian triangle of X(5), and define B'_5 and C'_5 cyclically. \
The circumcircles of AB'_5C'_5, BC'_5A'_5, CA'_5B'_5 concur in X(265).\

4) Let A_{F_P}B_{F_P}C_{F_P} be the X(3)-Fuhrmann triangle . \
Let A'' be the reflection of A in line B_{F_P}C_{F_P} , \
and define B'' and C'' cyclically. \
Then A''B''C'' is inversely similar to ABC, \
with similtude center X(265), \
and A''B''C'' is perspective to ABC with persepctor X(74).\

5) Let A_a, B_b, C_c be the reflection triangle. \
Let L_a be the line through A_a parallel to the Euler line, \
and define L_b and L_c cyclically. \
Let L'_a be the reflection of  L_a in sideline BC, \
and defineL'_b and L'_c cyclically. \
The lines L'_a, L'_b, L'_c concur in X(265).\

6) Let A_a, B_b, C_c   be the reflection triangle. \
Let A_p be the trilinear pole of line B_bC_c, \
and define B_p, C_p cyclically. \
The lines AA_p, BB_p, CC_p concur in X(5). \
Let A^* be the trilinear pole, wrt A_aB_bC_c, \
of line B_pC_p, and define B^*, C^* cyclically. \
The lines AA^*, BB^*, CC^* \textcolor{red}{concur in X(265) ???}.\

7) Let A_H be the point such that triangle A_HBC is \
directly similar to the orthic triangle, \
and define B_H, C_H cyclically. \
The lines AA_H BB_H, CC_H concur in X(265). \
If 'inversely' is substituted for 'directly', \
the lines AA'_H BB'_H, CC'_H concur in X(3).\

\vspace{1.5cm}
Geogebra\ contributed\ by\ René\ Drucker,\ june 2022.\ 
}}

Position(M,Séquence(TriangleCentre(A,B,C,n),n,1,5000))
X_{n}=TriangleCentre(A, B, C, n)

X_{5}

euler

Euler

35

33

20

L(2,6)

L(3,35)

L(1,4)

L(7,20)

55

L(11,55)

69

L(6,69)

L(7,9)

L(1,8)

L(3,9)

Brocard axe

droite OI

Euler

Van Aubel

L(4,9)

L(4,8)

S_p

L(5,13)

Napoléon

Nagel

S_{ch}

Napoleon cubic

\text{\bold{\textcolor{red}{X(265)\ }}\
8) Let Ai be the isogonal conjugate of A \
wrt the A-altimedial triangle, \
and define Bi and Ci cyclically. \
The lines AAi, BBi, CCi concur in X(265).\
9) Let La be the line through A parallel to the Euler line \
of the A-altimedial triangle, \
and define Lb and Lc cyclically. \
Lines La, Lb, Lc concur in X(265).\
10) Let O_A be the circle centered at the A-vertex of the \
orthocentroidal triangle and passing through A; \
define O_B and O_C cyclically. \
X(265) is the radical center of O_A, O_B, O_C.\
11) X(265) lies on the Jerabek circumhyperbola\
X(265) lies onthe curves K025\
X(265) lies on these lines:\
{3,125}\
{4,94}\
{5,49}\
{6,13}\
{30,74}\
12) X(265) = reflection of X(110) in X(5) . \
13) X(265) = isogonal conjugate of X(186)\
14) Let A^*B^*C^* be the Kosnita triangle. \
Let A' the orthopole of line B^*C^*, and define B' and C' cyclically.\
The lines AA', BB', CC' concur in X(265).\
\vspace{1.5cm}
Geogebra\ contributed\ by\ René\ Drucker,\ june 2022.\ 
}}

Position(M,Séquence(TriangleCentre(A,B,C,n),n,1,5000))
X_{n}=TriangleCentre(A, B, C, n)

Euler

Longchamps

Euler

$B^*$

$C^*$

$A^*$

Euler-A

Longchamps

Euler

Euler-B

Longchamps

Euler

Euler-C

Longchamps

Euler

Jerabek

cubique-025

IsogonalConjugate-186

$C^*$

$B^*$

$A^*$

A_{F_P}

B_{F_P}

C_{F_P}

A = (40.26022979375839, 59.3702782985408)
B = (-37.50764996426329, 0.5142616732317893)
C = (63.26975804148375, 1.8557415704449776)
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_{1} = Midpoint(A, H)
B_{1} = Midpoint(B, H)
h_B = Line(B, H_B)
h_C = Line(C, H_C)
N_9 = TriangleCenter(A, B, C, 5)
p = Line(H, O)
N_{ag} = TriangleCenter(A, B, C, 8)
q = Segment(A, Ti_A)
r = Segment(B, Ti_B)
s = Segment(C, Ti_C)
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 = Segment(A, A_e)
f_1 = Segment(C_e, C)
g_1 = Segment(B_e, B)
h_1 = Line(M_B, M_C)
i_1 = Line(M_C, M_A)
j_1 = Line(M_A, M_B)
k_1 = Line(Ti_B, Ti_C)
l_1 = Line(Ti_C, Ti_A)
m_1 = Line(Ti_A, Ti_B)
L = Intersect(j_1, m_1)
t_1 = Line(H, O)
b_1 = Segment(H_B, H_C)
c_1 = Segment(H_C, H_A)
d_1 = Segment(H_A, H_B)
e_1 = Segment(M_C, M_B)
f_2 = Segment(M_B, M_A)
g_2 = Segment(M_A, M_C)
h_2 = Line(H_B, H_C)
i_2 = Line(H_C, H_A)
j_2 = 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)
q_1 = Line(G, L_e)
a_1 = Line(O, X_{35})
k_2 = Line(H, X_{33})
m_2 = Line(G_e, X_{20})
b_{e_A} = OrthogonalLine(A, b_A)
b_{eB} = OrthogonalLine(B, b_B)
b_{e_C} = OrthogonalLine(C, b_C)
m_A = Line(A, M_A)
m_B = Line(B, M_B)
m_C = Line(C, M_C)
t1, m_a, m_c, m_b = Polygon(M_C, M_B, M_A)
X_{55} = TriangleCenter(A, B, C, 55)
f = Line(X_{55}, F_{eu})
X_{69} = TriangleCenter(A, B, C, 69)
g = Line(L_e, X_{69})
h = Line(G_e, M_{it})
i = Line(I, N_{ag})
j = LineBisector(B, C)
k = LineBisector(A, C)
l = LineBisector(B, A)
X_{84} = TriangleCenter(A, B, C, 84)
m_{AB} = Line(O, X_{84})
X_{15} = TriangleCenter(A, B, C, 15)
X_{16} = TriangleCenter(A, B, C, 16)
X_{32} = TriangleCenter(A, B, C, 32)
d = Line(O, X_{32})
X_{36} = TriangleCenter(A, B, C, 36)
X_{40} = TriangleCenter(A, B, C, 40)
X_{46} = TriangleCenter(A, B, C, 46)
e = Line(O, I)
L_{2,3} = Line(O, G)
h_A = Line(A, H_A)
p_2 = Line(H, L_e)
q_2 = Line(H, M_{it})
r_2 = Line(H, N_{ag})
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_2 = Circle(M_C, M_B, M_A)
C_{1} = 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_{2,6} = Line(G, X_{69})
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})
m = Mirror(m_A, b_A)
n_1 = Mirror(m_C, b_C)
l_2 = Mirror(m_B, b_B)
Ie_C = Intersect(b_{e_A}, b_{eB})
Ie_A = Intersect(b_{eB}, b_{e_C})
Ie_B = Intersect(b_{e_C}, b_{e_A})
Te_A = ClosestPoint(a, Ie_A)
Te_B = ClosestPoint(b, Ie_B)
Te_C = ClosestPoint(c, Ie_C)
n_2 = Segment(A, Te_A)
s_2 = Segment(B, Te_B)
t_2 = Segment(C, Te_C)
t2, te_a, te_b, te_c = Polygon(Te_B, Te_C, Te_A)
L_{1,2} = Line(N_{ag}, I)
d_2 = Circle(Ie_C, Te_C)
e_2 = Circle(Ie_B, Te_B)
c_3 = Circle(Ie_A, Te_A)
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 14, step 1)
poly1, a_2, b_2, f_3, S_C = Polygon(B, A, 3)
poly2, g_3, h_3, i_3, S_A = Polygon(C, B, 3)
poly3, j_3, k_3, l_3, S_B = Polygon(A, C, 3)
J_C = Centroid(poly1)
J_B = Centroid(poly3)
J_A = Centroid(poly2)
m_3 = Segment(A, J_A)
n_3 = Segment(B, J_B)
p_3 = Segment(C, J_C)
L_{2,17} = Line(G, X_{62})
L_{3,13} = Line(O, X_{13})
L_{4,15} = Line(H, X_{15})
poly4, q_3, r_3, s_3, D = Polygon(J_C, J_A, 3)
eq1 = Cubic(A, B, C, 5)
X_{265} = TriangleCenter(A, B, C, 265)
X_{125} = TriangleCenter(A, B, C, 125)
X_{74} = TriangleCenter(A, B, C, 74)
p_1 = Segment(O, X_{125})
r_1 = Segment(X_{125}, X_{265})
Ct, Bt, At, s_1, t_3, a_3, b_3, d_3, e_3 = triangleTangentiel(A, B, C)
A't = Mirror(At, a)
B't = Mirror(Bt, b)
C't = Mirror(Ct, a)
c_4 = Circle(C, A't, B)
d_4 = Circle(A, B't, C)
e_4 = Circle(A, B, C't)
B_5, C_5, A_5 = triangleQanticevian(A, B, C, N_9)
A'_5 = Mirror(A_5, a)
B'_5 = Mirror(B_5, b)
C'_5 = Mirror(C_5, c)
f_4 = Circle(A, B'_5, C'_5)
g_4 = Circle(B, A'_5, C'_5)
h_4 = Circle(C, B'_5, A'_5)
A_a = Mirror(A, a)
B_b = Mirror(B, b)
C_c = Mirror(C, c)
l_4, F, J, K, M, N, p_4 = droite2Euler(A, B, C)
L_a = Line(A_a, l_4)
L_b = Line(B_b, l_4)
L_c = Line(C_c, l_4)
L'_c = Mirror(L_c, c)
L'_b = Mirror(L_b, b)
L'_a = Mirror(L_a, a)
A_p = TrilinearPole(A, B, C, B_b, C_c)
B_p = TrilinearPole(A, B, C, A_a, C_c)
C_p = TrilinearPole(A, B, C, B_b, A_a)
q_4 = Line(A, N_9)
r_4 = Line(B, B_p)
s_4 = Line(C, C_p)
o: (A * TrilinearPole[A_a, B_b, C_c, B_p, C_p])
BB = TrilinearPole(A_a, B_b, C_c, A_p, C_p)
CC = TrilinearPole(A_a, B_b, C_c, A_p, B_p)
AA = TrilinearPole(A_a, B_b, C_c, B_p, C_p)
a'_{1} = Rotate(a, Angle[H_C, H_B, H_A], B)
a'_{2} = Rotate(a, Angle[H_B, H_C, H_A], C)
A_H = Intersect(a'_{2}, a'_{1})
t_4 = Line(A_H, A)
h_a = Angle(H_B, H_A, H_C)
h_b = Angle(H_C, H_B, H_A)
h_c = Angle(H_A, H_C, H_B)
b'_{2} = Rotate(b, (-h_a), A)
b'_{1} = Rotate(b, h_c, C)
B_H = Intersect(b'_{2}, b'_{1})
a_4 = Line(B, B_H)
c'_{2} = Rotate(c, h_a, A)
c'_{1} = Rotate(c, (-h_b), B)
C_H = Intersect(c'_{2}, c'_{1})
b_4 = Line(C, C_H)
A'_H = Mirror(A_H, a)
B'_H = Mirror(B_H, b)
C'_H = Mirror(C_H, c)
f_5 = Line(A, A'_H)
g_5 = Line(B, B'_H)
h_5 = Line(C, C'_H)
Ai, Ai_{2}, Ai_{3}, Ai_{4}, Ai_{5}, Ai_{6}, Ai_{7}, Ai_{8}, Ai_{9}, Ai_{10} = IsogonalConjugate(H_A, M_B, M_C, A)
Bi, Bi_{2}, Bi_{3}, Bi_{4}, Bi_{5}, Bi_{6}, Bi_{7}, Bi_{8}, Bi_{9}, Bi_{10} = IsogonalConjugate(H_B, M_A, M_C, B)
Ci, i_5, j_5, k_5, l_5, m_5, n_5, p_5, q_5, r_5 = IsogonalConjugate(H_C, M_B, M_A, C)
s_5 = Line(Ai, A)
t_5 = Line(Bi, B)
a_5 = Line(C, Ci)
b_5, P, Q, R, S, T, c_5 = droite2Euler(H_A, M_B, M_C)
d_5, U, V, W, Z, D_1, e_5 = droite2Euler(H_B, M_A, M_C)
f_6, E_1, F_1, G_1, H_1, I_1, c_6 = droite2Euler(H_C, M_B, M_A)
u = IndexOf(H_1, Sequence[TriangleCenter[A, B, C, n], n, 1, 5000])
g_6 = Line(A, b_5)
h_6 = Line(B, d_5)
i_6 = Line(C, f_6)
J_1, K_1, L_1, j_6, k_6, l_6 = TriangleOrthocentroidal(A, B, C)
d_6 = Circle(K_1, A)
e_6 = Circle(J_1, C)
p_6 = Circle(L_1, B)
M_1, N_1 = Intersect(e_6, d_6)
m_6 = Line(N_1, M_1)
q_6 = Conic(A, B, C, O, H)
eq2 = Cubic(A, B, C, 25)
X_{49} = TriangleCenter(A, B, C, 49)
X_{94} = TriangleCenter(A, B, C, 94)
n_6 = Line(O, X_{125})
r_6 = Line(H, X_{94})
s_6 = Line(X_{49}, N_9)
t_6 = Line(L_e, X_{13})
a_6 = Line(X_{74}, L_{2,3})
X_{110} = TriangleCenter(A, B, C, 110)
b_6 = Segment(X_{110}, N_9)
f_7 = Segment(N_9, X_{265})
X_{186} = TriangleCenter(A, B, C, 186)
O_1, g_7, h_7, i_7, j_7, k_7, l_7, m_7, n_7, p_7 = IsogonalConjugate(A, B, C, X_{186})
P_1, Q_1, R_1, q_7, r_7, s_7 = triangle2Kosnita(A, B, C)
A' = orthopôle(A, B, C, P_1, Q_1)
B' = orthopôle(A, B, C, P_1, R_1)
C' = orthopôle(A, B, C, R_1, Q_1)
t_7 = Line(A', A)
a_7 = Line(B, B')
b_7 = Line(C, C')
E, S_1, T_1 = Fuhrmann_P_trangle(A, B, C, O)
A'' = Mirror(A, Line[S_1, T_1])
B'' = Mirror(B, Line[E, T_1])
C'' = Mirror(C, Line[S_1, E])
i_4 = Line(B, X_{74})
j_4 = Line(C, X_{74})
k_4 = Line(A, A'')

Tool: triangleTangentiel
sommetsx3
Input: A, B, C
Output: C', B', A', g_2, h_2, f_2, r_1, t_1, e_1

O = TriangleCenter(A, B, C, 3)
r_1 = OrthogonalLine(A, Line[A, O])
t_1 = OrthogonalLine(B, Line[B, O])
e_1 = OrthogonalLine(C, Line[C, O])
C' = Intersect(t_1, r_1)
A' = Intersect(t_1, e_1)
B' = Intersect(r_1, e_1)
f_2 = Segment(C', B')
g_2 = Segment(B', A')
h_2 = Segment(A', C')

Tool: triangleAntiCevianOK
sommetsx3P
Input: A, B, C, P
Output: B', A', C'

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: triangleQanticevian
sommetsx3-Point
Input: A, B, C, Q
Output: B_Q, C_Q, A_Q

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
A'_Q = Intersect(a, Line[A, Q])
B'_Q = Intersect(b, Line[B, Q])
C'_Q = Intersect(c, Line[C, Q])
t_3 = Line(C'_Q, B'_Q)
a_3 = Line(B'_Q, A'_Q)
b_3 = Line(A'_Q, C'_Q)
A_0 = Intersect(a, t_3)
C_0 = Intersect(a_3, c)
B_0 = Intersect(b_3, b)
f_1 = Line(A_0, A)
g_1 = Line(C_0, C)
p_1 = Line(B_0, B)
C_Q = Intersect(f_1, p_1)
A_Q = Intersect(p_1, g_1)
B_Q = Intersect(g_1, f_1)

Tool: triangle2Furhmann
sommetsx3
Input: A, B, C
Output: t1, B', A', C', t, g_1, f_1

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
d = Circle(A, B, C)
A_0, E = Intersect(d, LineBisector[B, C])
F, B_0 = Intersect(d, LineBisector[A, C])
H_{1}, C_0 = Intersect(d, LineBisector[B, A])
A' = Mirror(A_0, a)
B' = Mirror(B_0, b)
C' = Mirror(C_0, c)
t1, t, f_1, g_1 = Polygon(A', C', B')

Tool: Triangle2Fuhrmann2
sommetsx3
Input: A, B, C
Output: C'', B'', A'', b'', c'', a''

a = Line(B, C)
b = Line(C, A)
c = Line(A, B)
c_O = Circle(A, B, C)
O = TriangleCenter(A, B, C, 3)
A_0, E = Intersect(c_O, LineBisector[B, C])
F, B_0 = Intersect(c_O, LineBisector[A, C])
H_{1}, C_0 = Intersect(c_O, LineBisector[B, A])
A'_0 = Mirror(A_0, O)
C'_0 = Mirror(C_0, O)
B'_0 = Mirror(B_0, O)
A'' = Mirror(A'_0, a)
B'' = Mirror(B'_0, b)
C'' = Mirror(C'_0, c)
t3, c'', a'', b'' = Polygon(A'', B'', C'')

Tool: droite2Euler
sommetsx3
Input: C, B, A
Output: m_1, L_o, G, H, O, N_9, c_2

Euler

Euler

Longchamps

a = Line(C, B)
H = TriangleCenter(A, B, C, 4)
O = TriangleCenter(A, B, C, 3)
N_9 = TriangleCenter(A, B, C, 5)
m_1 = Line(O, H)
b = Line(A, C)
c = Line(B, A)
A_H = ClosestPoint(a, A)
B_H = ClosestPoint(b, B)
C_H = ClosestPoint(c, C)
c_2 = Circle(A_H, B_H, C_H)
G = TriangleCenter(A, B, C, 2)
L_o = TriangleCenter(A, B, C, 20)

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: TrilinearPole
sommetsx3DroitePointPoint
Input: A, B, C, M, N
Output: T

a = Line(B, C)
c = Line(A, B)
l = Line(M, N)
A_p = Intersect(l, a)
A'_p = ConjuguéHarmonique(B, C, A_p)
C_p = Intersect(c, l)
C'_p = ConjuguéHarmonique(A, B, C_p)
d = Line(C'_p, C)
e = Line(A'_p, A)
T = Intersect(e, d)

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: TriangleOrthocentroidal
sommetsx3
Input: A, B, C
Output: C', A', B', r_1, t_3, s_1

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)
H = TriangleCenter(A, B, C, 4)
X_{381} = TriangleCenter(A, B, C, 381)
A' = Mirror(H, OrthogonalLine[X_{381}, Line[H_A, H]])
B' = Mirror(H, OrthogonalLine[X_{381}, Line[H_B, H]])
C' = Mirror(H, OrthogonalLine[X_{381}, Line[H_C, H]])
t3, r_1, s_1, t_3 = Polygon(B', A', C')

Tool: triangle2Kosnita
sommetsx3
Input: A, B, C
Output: K_C, K_B, K_A, k_c, k_a, k_b

O = TriangleCenter(A, B, C, 3)
d = Circle(A, O, B)
e = Circle(B, O, C)
f = Circle(C, O, A)
K_C = Center(d)
K_B = Center(f)
K_A = Center(e)
t1, k_a, k_b, k_c = Polygon(K_B, K_C, K_A)

Tool: orthopôle
triangleABC,droiteMN
Input: A, B, C, M, N
Output: Ω

l = Line(N, M)
b = Line(C, A)
c = Line(A, B)
B' = ClosestPoint(l, B)
C' = ClosestPoint(l, C)
k = OrthogonalLine(B', b)
p = OrthogonalLine(C', c)
Ω = Intersect(p, k)

Tool: Fuhrmann_P_trangle
sommetsX3, point
Input: A, B, C, P
Output: A_0, B_0, C_0

A_{F_P}

B_{F_P}

C_{F_P}

O = TriangleCenter(C, B, A, 3)
c' = Line(B, A)
b' = Line(A, C)
a' = Line(C, B)
P_A = Mirror(A, OrthogonalLine[O, Line[P, A]])
P_B = Mirror(B, OrthogonalLine[O, Line[P, B]])
P_C = Mirror(C, OrthogonalLine[O, Line[P, C]])
A_0 = Mirror(P_A, a')
B_0 = Mirror(P_B, b')
C_0 = Mirror(P_C, c')