X(12)    HARMONIC CONJUGATE OF X(11) 
with respect to X(1) and X(5)

Here, X(12), labeled 12, is constructed 
as the point of intersection of lines 
X(1)X(5) and X(2)X(56).

Sketch contributed by Michael Stairs, 
January 2018.

Drag A, B, C.

8

56

1

2

3

4

5

11

12

A = (-1.5129514408959661, 3.688862799722028)
B = (-1.0631975401137612, 2.0727116930966556)
C = (0.13003720326105656, 2.0727116930966556)
f = Segment(B, A)
g = Segment(A, C)
h = Segment(C, B)
i = AngularBisector(B, C, A)
j = AngularBisector(C, B, A)
C'_1 = Midpoint(f)
B'_1 = Midpoint(g)
A'_1 = Midpoint(h)
m = OrthogonalLine(A'_1, h)
D_1 = Point(m)
k = OrthogonalLine(A, h)
l = OrthogonalLine(C, f)
n = OrthogonalLine(B, g)
p = LineBisector(h)
q = LineBisector(g)
r = LineBisector(f)
b = AngularBisector(B, A, C)
c = AngularBisector(A, B, C)
d = AngularBisector(A, C, B)
D = Intersect(d, f)
E = Intersect(c, g)
F = Intersect(b, h)
s = AngularBisector(B, A, C)
t = AngularBisector(A, B, C)
a = AngularBisector(A, C, B)
e = Line(A, C)
f_1 = Line(C, B)
g_1 = Line(B, A)
G = Point(e)
H = Point(g_1)
I = Point(f_1)
h_1 = AngularBisector(I, B, A)
i_1 = AngularBisector(H, A, C)
j_1 = AngularBisector(G, C, B)
J = Intersect(h_1, i_1)
K = Intersect(i_1, j_1)
L = Intersect(h_1, j_1)
k_1 = Segment(J, K)
l_1 = Segment(K, L)
m_1 = Segment(L, J)
n_1 = OrthogonalLine(L, h)
p_1 = OrthogonalLine(K, g)
q_1 = OrthogonalLine(J, f)
C' = Intersect(g_1, q_1)
B' = Intersect(e, p_1)
A' = Intersect(f_1, n_1)
c_1 = Circle(L, A')
d_1 = Circle(K, B')
e_1 = Circle(J, C')
r_1 = Line(A, A')
s_1 = Line(B, B')
t_1 = Line(C, C')
M = Intersect(r_1, s_1)
a_1 = Segment(C, C')
b_1 = Segment(B, B')
f_2 = Segment(A', A)
g_2 = AngularBisector(B, A, C)
h_2 = AngularBisector(A, B, C)
i_2 = AngularBisector(A, C, B)
C'_2 = Mirror(C, i_2)
C'' = Mirror(C', i_2)
j_2 = Segment(C'_2, C'')
A'' = Mirror(A', g_2)
A'_2 = Mirror(A, g_2)
k_2 = Segment(A'', A'_2)
B'_2 = Mirror(B, h_2)
B'' = Mirror(B', h_2)
l_2 = Segment(B'_2, B'')
N = Intersect(j_2, k_2)
O = IncenterX(A, B, C)
P = Midpoint(A, B)
Q = Midpoint(B, C)
R = Midpoint(C, A)
m_2 = Segment(B, R)
n_2 = Segment(A, Q)
p_2 = Segment(C, P)
S = Intersect(m_2, n_2)
q_2 = LineBisector(f)
r_2 = LineBisector(g)
s_2 = LineBisector(h)
T = Intersect(q_2, r_2)
t_2 = OrthogonalLine(A, h)
a_2 = OrthogonalLine(B, g)
b_2 = OrthogonalLine(C, f)
U = Intersect(t_2, a_2)
V = Midpoint(U, T)
c_2 = Circle(V, R)
d_2 = Incircle(A, B, C)
W = Intersect(c_2, d_2, 1)
e_2 = OrthogonalLine(A, h)
f_3 = OrthogonalLine(B, g)
g_3 = OrthogonalLine(C, f)
h_3 = Line(O, V)
i_3 = Line(S, N)
Z = Intersect(h_3, i_3)

Tool: Incenter X(1) (command IncenterX)
Incenter[ <Point>, <Point>, <Point> ]
Input: A, B, C
Output: I

s = AngularBisector(B, A, C)
t = AngularBisector(A, B, C)
I = Intersect(s, t)

Tool: Incircle
Input: A, B, C
Output: c_1

g = Segment(A, C)
s = AngularBisector(B, A, C)
t = AngularBisector(A, B, C)
I = Intersect(s, t)
e = OrthogonalLine(I, g)
G = Intersect(e, g)
c_1 = Circle(I, G)

Tool: Exsimilicenter X(56) (command Tool4)
Input: A, B, C
Output: N

56

f = Segment(B, A)
h = Segment(C, B)
e = Line(A, C)
f_1 = Line(C, B)
g_1 = Line(B, A)
G = Point(e)
H = Point(g_1)
I = Point(f_1)
h_1 = AngularBisector(I, B, A)
i_1 = AngularBisector(H, A, C)
j_1 = AngularBisector(G, C, B)
J = Intersect(h_1, i_1)
L = Intersect(h_1, j_1)
n_1 = OrthogonalLine(L, h)
q_1 = OrthogonalLine(J, f)
C' = Intersect(g_1, q_1)
A' = Intersect(f_1, n_1)
g_2 = AngularBisector(B, A, C)
i_2 = AngularBisector(A, C, B)
C'_2 = Mirror(C, i_2)
C'' = Mirror(C', i_2)
j_2 = Segment(C'_2, C'')
A'' = Mirror(A', g_2)
A'_2 = Mirror(A, g_2)
k_2 = Segment(A'', A'_2)
N = Intersect(j_2, k_2)

Tool: Ninepoint Center X(5) (command Tool5)
Tool5[ <Point>, <Point>, <Point> ]
Input: A, B, C
Output: V

3

4

5

f = Segment(B, A)
g = Segment(A, C)
h = Segment(C, B)
q_2 = LineBisector(f)
r_2 = LineBisector(g)
T = Intersect(q_2, r_2)
t_2 = OrthogonalLine(A, h)
a_2 = OrthogonalLine(B, g)
U = Intersect(t_2, a_2)
V = Midpoint(U, T)