Basic Parameters of the Kimberling Center X(42)

​
classification: odd center
standard barycenter:
2
a
(2s-a)
2s(
2
r
-2rR+
2
s
)

AX
42
 = 9.90168
bc(bc+2rR)
2
r
-2rR+
2
s
-
4
2
r
2
R
(
2
r
+4rR+5
2
s
)
2
(
2
r
-2rR+
2
s
)
= 9.90168
d
a
= 8.14378
ar(2s-a)
2
r
-2rR+
2
s
= 8.14378
d
X
42
= 21.7671
2r(
2
r
+4rR+
2
s
)
2
r
-2rR+
2
s
= 21.7671
Given a triangle
ABC
, the Kimberling center
X
42
is the crosspoint of the incenter
X
1
and the symmedian point
X
6
[1].
X
42
lies on the incenter-centroid line
X
1
X
2
.
Let
a
,
b
,
c
be the side lengths
BC,CA,AB
,
R
,
r
,
s
be the circumradius, inradius and semiperimeter of
ABC
,
S=2ABC
,
d
a
,
d
b
,
d
c
be the exact trilinear coordinates of
X
42
with respect to
ABC
and
d
X
42
=
d
a
+
d
b
+
d
c
.
Then

AX
42
=
bc(bc+2Rr)
2
r
-2Rr+
2
s
-
4
2
R
2
r
(
2
r
+4Rr+5
2
s
)
2
(
2
r
-2Rr+
2
s
)
,
d
a
=
ar(2s-a)
2
r
-2rR+
2
s
,
d
X
42
=
2r(
2
r
+4rR+
2
s
)
2
r
-2rR+
2
s
.
You can drag the vertices
A
,
B
,
C
.

Details

A triangle center is said to be "even center" if 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 barycentric coordinates as a constant, it is called a "neutral center" (The centroid
X
2
is the only "neutral center"). Conversely, a triangle center is said to be "odd center" if it is neither even nor neutral.
Standard barycentric coordinates of a point with respect to a reference triangle have a sum of 1.
Classification: odd center

References

[1] C. Kimberling. "Encyclopedia of Triangle Centers."
faculty.evansville.edu/ck6/encyclopedia.

External Links

Incenter (Wolfram MathWorld)
Triangle Centroid (Wolfram MathWorld)
Symmedian Point (Wolfram MathWorld)
Crosspoint (Wolfram MathWorld)
Relating Trilinear and Tripolar Coordinates for a Triangle

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Kimberling Center X(42)"​
​http://demonstrations.wolfram.com/BasicParametersOfTheKimberlingCenterX42/​
​Wolfram Demonstrations Project​
​Published: November 21, 2022