X(11)    FEUERBACH POINT

The Feuerbach point, labeled 11, is the touchpoint 
of the incircle and the nine-point circle.

Also shown are the vertices of the Feuerbach 
triangle, here denoted by A',B', and C'.

Sketch contributed by Michael Stairs, January 2018.

Drag A, B, C.

1

5

11

A = (-0.419433562014221, 5.708003844116602)
B = (0.006183349173530931, 2.0307034593049345)
C = (2.670427653283165, 2.0595000340762644)
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)
D = Midpoint(A, C)
E = Midpoint(C, B)
F = Midpoint(B, A)
s = Segment(F, D)
t = Segment(D, E)
a = Segment(E, F)
b = Line(B, A)
c = Line(A, C)
d = Line(C, B)
G = Point(b)
H = Point(c)
I = Point(d)
e = AngularBisector(I, B, A)
f_1 = AngularBisector(G, A, C)
g_1 = AngularBisector(H, C, B)
J = Intersect(e, f_1)
K = Intersect(f_1, g_1)
L = Intersect(e, g_1)
h_1 = Segment(J, L)
i_1 = Segment(L, K)
j_1 = Segment(K, J)
k_1 = OrthogonalLine(J, f)
l_1 = OrthogonalLine(K, g)
m_1 = OrthogonalLine(L, h)
C'_2 = Intersect(b, k_1)
c_1 = Circle(J, C'_2)
N = Intersect(d, m_1)
e_1 = Circle(L, N)
B'_2 = Intersect(c, l_1)
d_1 = Circle(K, B'_2)
p_1 = NinepointCircle(A, B, C)
A' = Intersect(e_1, p_1, 1)
C' = Intersect(c_1, p_1, 1)
B' = Intersect(d_1, p_1, 1)
n_1 = AngularBisector(B, A, C)
q_1 = AngularBisector(A, B, C)
M = Intersect(n_1, q_1)
O = Center(p_1)
r_1 = OrthogonalLine(M, g)
P = Intersect(c, r_1)
s_1 = Circle(M, P)
Q = Intersect(p_1, s_1, 1)

Tool: Ninepoint Circle (command NinepointCircle)
Input: A, B, C
Output: c_1

4

3

5

f = Segment(B, A)
g = Segment(A, C)
h = Segment(C, B)
k = OrthogonalLine(A, h)
l = OrthogonalLine(C, f)
E_1 = Intersect(k, l)
p = LineBisector(h)
q = LineBisector(g)
F_1 = Intersect(p, q)
E = Midpoint(C, A)
G = Midpoint(F_1, E_1)
c_1 = Circle(G, E)