X(7)    GERGONNE POINT

The Gergonne point, labeled 7, is the 
point of concurrence of lines AA', BB', 
CC', where A', B',C' are the touchpoints 
of sides BC, CA, AB with the incircle.

Sketch contributed by Michael Stairs, 
January 2018.

Drag A, B, C.

1

7

A = (0.11074014111712231, 3.328352243856815)
B = (-0.5384617004849285, 1.639430632208674)
C = (3.303716376865234, 1.6446243381207863)
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)
I = Intersect(s, t)
e = OrthogonalLine(I, g)
B' = Intersect(e, g)
c_1 = Circle(I, B')
f_1 = OrthogonalLine(I, f)
g_1 = OrthogonalLine(I, h)
C' = Intersect(c_1, f_1, 2)
A' = Intersect(c_1, g_1, 2)
h_1 = Segment(B, B')
i_1 = Segment(A, A')
j_1 = Segment(C, C')
G = Intersect(h_1, i_1)
k_1 = Segment(B, I)
l_1 = Segment(A, I)
m_1 = Segment(C, I)
n_1 = Segment(I, B')
p_1 = Segment(I, C')
q_1 = Segment(I, A')

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)