X(27)    CEVAPOINT OF X(4) AND X(19)

The cevapont of X(4) and X(19), labeled 27, 
is constructed here as the point of intersection 
of lines X(2)X(3) and X(7)X(81).

Sketch contributed by Michael Stairs, March 2018.

Drag A, B, C.

2

6

81

3

2

7

2

27

A = (2.2850524857915517, 5.758675882008473)
B = (-1.3477622719484024, 0.27039122112164726)
C_1 = (3.147683510085133, 0.29354448113962284)
f = Segment(B, A)
g = Segment(A, C_1)
h = Segment(C_1, B)
i = AngularBisector(B, C_1, A)
j = AngularBisector(C_1, 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_1, f)
n = OrthogonalLine(B, g)
p = LineBisector(h)
q = LineBisector(g)
r = LineBisector(f)
b = AngularBisector(B, A, C_1)
c = AngularBisector(A, B, C_1)
d = AngularBisector(A, C_1, B)
D = Intersect(d, f)
E_1 = Intersect(c, g)
F = Intersect(b, h)
s = AngularBisector(B, A, C_1)
t = AngularBisector(A, B, C_1)
a = AngularBisector(A, C_1, B)
G = Incenter(A, B, C_1)
e = Segment(A, G)
f_1 = Segment(B, G)
g_1 = Segment(G, C_1)
triangle1, a_1, b_1, g_2 = Polygon(B, G, A)
triangle2, g_3, b_2, c_1 = Polygon(B, C_1, G)
h_1 = Line(A, C_1)
i_1 = Line(C_1, B)
j_1 = Line(B, A)
k_1 = OrthogonalLine(A, g_3)
l_1 = OrthogonalLine(B, g)
m_1 = OrthogonalLine(C_1, g_2)
H = Intersect(k_1, l_1)
I = Centroid(A, B, C_1)
E = Line(I, H)
J = Centroid(A, G, C_1)
K = Orthocenter(A, G, C_1)
E_B = Line(K, J)
L = Orthocenter(A, G, B)
M = Centroid(A, G, B)
E_C = Line(M, L)
N = Orthocenter(B, G, C_1)
O = Centroid(G, B, C_1)
E_A = Line(O, N)
P = Intersect(E, E_B)
n_1 = Segment(A, B)
p_1 = Segment(B, C_1)
Q = Midpoint(n_1)
S = Midpoint(g)
q_1 = Segment(B, S)
r_1 = Segment(Q, C_1)
R = Intersect(q_1, r_1)
s_1 = AngularBisector(A, B, C_1)
t_1 = AngularBisector(A, C_1, B)
Q' = Mirror(Q, t_1)
C = Mirror(C_1, t_1)
d_1 = Segment(Q', C)
B' = Mirror(B, s_1)
S' = Mirror(S, s_1)
e_1 = Segment(B', S')
T = Intersect(d_1, e_1)
f_2 = Line(P, G)
h_2 = Line(R, T)
U = Intersect(f_2, h_2)
i_2 = LineBisector(n_1)
j_2 = LineBisector(g)
V = Intersect(i_2, j_2)
k_2 = AngularBisector(A, B, C)
l_2 = AngularBisector(A, C, B)
W = Intersect(k_2, l_2)
n_2 = AngularBisector(A, B, C)
p_2 = AngularBisector(A, C, B)
q_2 = OrthogonalLine(W, p_1)
Z = Intersect(q_2, p_1)
c_2 = Circle(W, Z)
r_2 = Segment(A, Z)
A_1 = Intersect(c_2, g, 1)
s_2 = Segment(B, A_1)
B_1 = Intersect(r_2, s_2)
t_2 = Line(U, B_1)
F_1 = Midpoint(p_1)
G_1 = Midpoint(n_1)
m_2 = Segment(A, F_1)
a_2 = Segment(C, G_1)
H_1 = Intersect(m_2, a_2)
d_2 = Line(V, H_1)
I_1 = Intersect(t_2, d_2)

Tool: Centroid
Input: A, B, C
Output: G

D = Midpoint(A, B)
E = Midpoint(B, C)
s = Segment(C, D)
t = Segment(A, E)
G = Intersect(s, t)

Tool: Incenter
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: Orthocenter
Input: A, B, C
Output: H

g = Segment(A, C)
G = Incenter(A, B, C)
triangle2, g_3, b_2, c_1 = Polygon(B, C, G)
k_1 = OrthogonalLine(A, g_3)
l_1 = OrthogonalLine(B, g)
H = Intersect(k_1, l_1)