X(5)    NINE-POINT CENTER

The nine-point center, labeled 5, is the center
of the nine-point circle.  X(5) is the midpoint
of X(3) and X(4).

The nine-point circle is the circumcircle of 
the medial triangle, DEF.  It is also the 
circumcircle of the orthic triangle, A'B'C'.

Sketch contributed by Michael Stairs, January 2018.

Drag A, B, C. 

4

3

5

A = (2.573066658291516, 4.575851334430599)
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)
E_1 = Intersect(k, l)
p = LineBisector(h)
q = LineBisector(g)
r = LineBisector(f)
F_1 = Intersect(p, q)
s = Ray(B, E_1)
t = Ray(A, E_1)
a = Ray(C, E_1)
A' = Intersect(h, t)
B' = Intersect(g, s)
b = Segment(A', B')
C' = Intersect(f, a)
c = Segment(B', C')
d = Segment(C', A')
D = Midpoint(B, C)
E = Midpoint(C, A)
F = Midpoint(A, B)
e = Segment(F_1, E_1)
f_1 = Segment(F, D)
g_1 = Segment(D, E)
h_1 = Segment(E, F)
G = Midpoint(F_1, E_1)
c_1 = Circle(G, E)