Basic Parameters of the Kimberling Center X12

​
classification: odd center
standard barycenter:
r
a
2
(2s-a)
4
2
s
(2r+R)

AX
12
 = 10.4778
2
a
(R-r)-2aRs
2r+R
+
2
s
(r+R)(2r+R)-
3
r
(2r+9R)
2
(2r+R)
= 10.4778
d
a
= 7.97923
bc
r
a
2
(2s-a)
8R
2
s
(2r+R)
= 7.97923
d
X
12
= 22.4562
r(
2
r
+2rR+4
2
R
+
2
s
)
2R(2r+R)
= 22.4562
Given a triangle
ABC
, let
A
',
B'
,
C'
be the points of tangency of the nine-point circle with the excircles opposite
A
,
B
,
C
. Then the lines
AA
',
BB'
,
CC'
are concurrent and the point of intersection is defined to be the Kimberling center
X
12
[1].
Let
a
,
b
,
c
be the side lengths,
R
,
r
,
s
be the circumradius, inradius and semiperimeter of
ABC
,
r
a
,
r
b
,
r
c
be the exradius,
S=2ABC
,
d
a
,
d
b
,
d
c
be the exact trilinear coordinates of
X
12
with respect to
ABC
and
d
X
12
=
d
a
+
d
b
+
d
c
.
Then

AX
12
=
2
a
(R-r)-2Rsa
R+2r
+
(R+r)(R+2r)
2
s
-
3
r
(9R+2r)
2
(R+2r)
,
d
a
=
bc
2
(2s-a)
r
a
8R(R+2r)
2
s
,
d
X
12
=
r(4
2
R
+
2
r
+2Rr+
2
s
)
2R(R+2r)
.
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." (Oct 13, 2022) faculty.evansville.edu/ck6/encyclopedia.

External Links

Nine-Point Circle (Wolfram MathWorld)
Excircles (Wolfram MathWorld)
Exspheres
Relating Trilinear and Tripolar Coordinates for a Triangle

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Kimberling Center X12"​
​http://demonstrations.wolfram.com/BasicParametersOfTheKimberlingCenterX12/​
​Wolfram Demonstrations Project​
​Published: October 25, 2022