X(13)    1st ISOGONIC CENTER (FERMAT POINT)

The Fermat Point, labeled 13, constructed 
from exterior equilateral triangles BCA', CAB', ABC'.

If ABC is an acute triangle, then X(13) is a point X 
for which angles BXC, CXA, AXB are equal.

Sketch contributed by Michael Stairs, January 2018.

Drag A, B, C.

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A = (0.5908268847290287, 3.3636925869876384)
B = (-1.0631975401137612, 2.0727116930966556)
C = (1.187057934835096, 2.1054030559288424)
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'_3 = Intersect(g_1, q_1)
B'_3 = Intersect(e, p_1)
A'_3 = Intersect(f_1, n_1)
c_1 = Circle(L, A'_3)
d_1 = Circle(K, B'_3)
e_1 = Circle(J, C'_3)
r_1 = Line(A, A'_3)
s_1 = Line(B, B'_3)
t_1 = Line(C, C'_3)
M = Intersect(r_1, s_1)
a_1 = Segment(C, C'_3)
b_1 = Segment(B, B'_3)
f_2 = Segment(A'_3, 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'_3, i_2)
j_2 = Segment(C'_2, C'')
A'' = Mirror(A'_3, g_2)
A'_2 = Mirror(A, g_2)
k_2 = Segment(A'', A'_2)
B'_2 = Mirror(B, h_2)
B'' = Mirror(B'_3, h_2)
l_2 = Segment(B'_2, B'')
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)
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)
e_2 = OrthogonalLine(A, h)
f_3 = OrthogonalLine(B, g)
g_3 = OrthogonalLine(C, f)
poly1, c_2, d_2, h_3, C' = Polygon(B, A, 3)
poly2, i_3, j_3, k_3, B' = Polygon(A, C, 3)
poly3, l_3, m_3, n_3, S = Polygon(B, C, 3)
poly3' = Mirror(poly3, h)
A' = Mirror(S, h)
p_3 = Segment(B, B')
q_3 = Segment(C, C')
r_3 = Segment(A, A')
N = Intersect(p_3, q_3)
s_3 = Line(B, B')
t_3 = Line(C, C')
a_3 = Line(A, A')

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

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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

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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)