X(6)    SYMMEDIAN POINT

The symmedian point, labeled 6, is the isogonal 
conjugate of X(2).

The lines AX(6), BX(6), CX(6), called symmedians, 
are reflections of the medians in the angle 
bisectors.

Sketch contributed by Michael Stairs, January 2018.

Drag A, B, C.

2

6

A = (2.7939742963877423, 4.202843877385203)
B = (-0.5384617004849285, 1.639430632208674)
C = (3.3139067437144787, 1.7202495506184519)
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)
C' = Midpoint(A, B)
A' = Midpoint(B, C)
B' = Midpoint(A, C)
s = Segment(C, C')
t = Segment(A, A')
a = Segment(B, B')
G = Intersect(s, t)
b = AngularBisector(B, A, C)
c = AngularBisector(A, B, C)
d = AngularBisector(A, C, B)
D = Intersect(d, f)
e = Segment(C, D)
E = Intersect(c, g)
f_1 = Segment(B, E)
F = Intersect(b, h)
g_1 = Segment(A, F)
C'_2 = Mirror(C, e)
C'' = Mirror(C', e)
s' = Segment(C'_2, C'')
B'_2 = Mirror(B, f_1)
B'' = Mirror(B', f_1)
a' = Segment(B'_2, B'')
A'_2 = Mirror(A, g_1)
A'' = Mirror(A', g_1)
t' = Segment(A'_2, A'')
H = Intersect(s', a')
h_1 = Segment(A, H)
i_1 = Segment(B, H)
j_1 = Segment(C, H)

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)