Basic Parameters of the Kimberling Center X33

​
classification: odd center
standard barycenter:
aS
S
ω
-4
2
R
+2R
S
B
S
C
2sS2rR-4
2
R
+
S
ω


AX
33
 = 3.75226
2rR
S
A
+bc
S
ω
-4
2
R

S
ω
-2R(2R-r)
-
2rR
S
ω
-4
2
R
-(2R-r)(r+4R)+
2
s
+
S
ω

2
(
S
ω
-2R(2R-r))
= 3.75226
d
a
= 9.25622
2
S
4
2
R
-
S
ω
-bc
S
B
S
C
2sS-2rR+4
2
R
-
S
ω

= 9.25622
d
X
33
= 14.2177
2r(
2
r
+6rR+8
2
R
-2
2
s
)
2
r
+2rR+4
2
R
-
2
s
= 14.2177
Given a triangle
ABC
, the Kimberling center
X
33
is the perspector of the orthic triangle (orange) and the intangents triangle (purple) of
ABC
[1].
Let
a
,
b
,
c
be the side lengths,
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
33
with respect to
ABC
and
d
X
33
=
d
a
+
d
b
+
d
c
.
Introduce the parameters
S
A
,
S
B
,
S
C
,
S
ω
in Conway notation, where
ω
is the Brocard angle.
Then

AX
33
=
2Rr
S
A
+bc
S
ω
-4
2
R

S
ω
-2R(2R-r)
-
2Rr
S
ω
-4
2
R

S
ω
+
2
s
-(4R+r)(2R-r)
2
(
S
ω
-2R(2R-r))
,
d
a
=
2
S
4
2
R
-
S
ω
-bc
S
B
S
C
2sS4
2
R
-
S
ω
-2Rr
,
d
X
33
=
2r(
2
r
+6rR+8
2
R
-2
2
s
)
2
r
+2rR+4
2
R
-
2
s
.
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

Orthic Triangle (Wolfram MathWorld)
Intangents Triangle (Wolfram MathWorld)
Perspector (Wolfram MathWorld)
Relating Trilinear and Tripolar Coordinates for a Triangle

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Kimberling Center X33"​
​http://demonstrations.wolfram.com/BasicParametersOfTheKimberlingCenterX33/​
​Wolfram Demonstrations Project​
​Published: October 25, 2022