Basic Parameters of the Kimberling Center X(44)

​
classification: odd center
standard barycenter:
a(2s-3a)
2(3r(r+4R)-
2
s
)

AX
44
 = 6.61384
2
a
(3
2
r
+12rR+
2
s
)-2as(3
2
r
+6rR+
2
s
)
2
s
-3r(r+4R)
+
-9
3
r
3
(r+4R)
-9
2
r
2
s
(
2
r
-4
2
R
)+r
4
s
(r-12R)+
6
s
2
(
2
s
-3r(r+4R))
= 6.61384
d
a
= 6.69813
rs(2s-3a)
3r(r+4R)-
2
s
= 6.69813
d
X
44
= 0
0
Given a triangle
ABC
, the Kimberling center
X
44
is the intersection of the lines
X
1
X
6
(incenter-symmedian point) and
X
2
X
89
(centroid-Kimberling center
X
89
)[1].
The Kimberling center
X
89
is the isogonal conjugate of
X
45
[1].
Let
a
,
b
,
c
be the side lengths,
R
,
r
,
s
be the circumradius, inradius and semiperimeter of
ABC
,
d
a
,
d
b
,
d
c
be the exact trilinear coordinates of
X
44
with respect to
ABC
and
d
X
44
=
d
a
+
d
b
+
d
c
.
Then

AX
44
=
2
a
(3
2
r
+12rR+
2
s
)-2as(3
2
r
+6rR+
2
s
)
2
s
-3r(4R+r)
+
-9
3
r
3
(r+4R)
-9
2
r
(
2
r
-4
2
R
)
2
s
+r(r-12R)
4
s
+
6
s
2
(-3r(r+4R)+
2
s
)
,
d
a
=
rs(2s-3a)
3r(4R+r)-
2
s
,
d
X
44
=0
.
You can drag the vertices
A
,
B
and
C
.

Details

A triangle center is said to be even when its barycentric coordinates can be expressed as a function of three variables
a
,
b
,
c
that all occur with even exponents. If the center of a triangle has constant barycentric coordinates, it is called a neutral center (the centroid
X
2
is the only neutral center). A triangle center is said to be odd if it is neither even nor neutral.
Standard barycentric coordinates of a point with respect to a reference triangle have a sum of 1.

References

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Nov 17, 2022) faculty.evansville.edu/ck6/encyclopedia.

External Links

Incenter (Wolfram MathWorld)
Symmedian Point (Wolfram MathWorld)
Triangle Centroid (Wolfram MathWorld)
Isogonal Conjugate (Wolfram MathWorld)
Relating Trilinear and Tripolar Coordinates for a Triangle
Basic Parameters of the Symmedian Point
Isogonal and Isotomic Conjugates

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Kimberling Center X(44)"​
​http://demonstrations.wolfram.com/BasicParametersOfTheKimberlingCenterX44/​
​Wolfram Demonstrations Project​
​Published: November 28, 2022