Basic Parameters of the Kimberling Center X(55)

​
classification: odd center
standard barycenter:
2
a
(s-a)
4rs(r+R)

AX
55
 = 8.51179
R(bc(r+R)-rR(r+4R))
r+R
= 8.51179
d
a
= 3.25293
a(s-a)
2(r+R)
= 3.25293
d
X
55
= 15.8865
r(r+4R)
r+R
= 15.8865
Given a triangle
ABC
, the Kimberling center
X
55
is the center of homothety of the tangential, intangents and extangents triangles[1].
The sides of the tangential triangle are tangent to the circumcircle of
ABC
at
A
,
B
,
C
. See the related links for the definitions of the intangents and extangents triangles.
The point
X
55
is on the line
IO
, where
I=
X
1
and
O=
X
3
are the incenter and circumcenter of
ABC
.
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
55
with respect to
ABC
and
d
X
55
=
d
a
+
d
b
+
d
c
.
Then

AX
55
=
R(bc(R+r)-Rr(4R+r))
R+r
,
d
a
=
a(s-a)
2(R+r)
,
d
X
55
=
r(4R+r)
R+r
.
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." (Jul 3, 2023) faculty.evansville.edu/ck6/encyclopedia.

External Links

Tangential Triangle (Wolfram MathWorld)
Intangents Triangle (Wolfram MathWorld)
Extangents Triangle (Wolfram MathWorld)
Homothetic Center (Wolfram MathWorld)
SubTriangle
TrianglePointData
Basic Parameters of the Kimberling Center X(51)
Basic Parameters of the Kimberling Center X(50)
Basic Parameters of the Kimberling Center X(49)
Basic Parameters of the Kimberling Center X(48)
Basic Parameters of the Kimberling Center X(47)
Basic Parameters of the Kimberling Center X(46)
Basic Parameters of the Kimberling Center X(45)
Basic Parameters of the Kimberling Center X(44)
Basic Parameters of the Kimberling Center X(42)
Basic Parameters of the Kimberling Center X(41)

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Kimberling Center X(55)"​
​http://demonstrations.wolfram.com/BasicParametersOfTheKimberlingCenterX55/​
​Wolfram Demonstrations Project​
​Published: July 19, 2023