Basic Parameters of the Triangle Centroid

​
classification: neutral center
standard barycenter:
1
3

AX
2
 = 2.53859
1
3
2(
2
b
+
2
c
)-
2
a
= 2.53859
d
a
= 1.16667
S
3a
=
2r
r
a
3(
r
a
-r)
= 1.16667
d
X
2
= 4.23284
2
r
+4rR+
2
s
6R
= 4.23284
Given a triangle
ABC
, let
M
A
,
M
B
,
M
C
be the midpoints of the sides
BC
,
CA
,
AB
. Then the three lines
AM
A
,
BM
B
,
CM
C
are called the medians and they intersect at a point
X
2
called the centroid of
ABC
[1].
Let
d
a
,
d
b
,
d
c
be the exact trilinear coordinates of
X
2
and let
d
X
1
=
d
A
+
d
B
+
d
C
.
Let
a
,
b
,
c
be the side lengths opposite the corresponding vertices of
ABC
and let
R
,
r
,
r
a
,
s
be the circumradius, inradius, exradius for
A
and semiperimeter of
ABC
.
Let
S=2ABC
and let
A
1
be the foot of the perpendicular from
X
2
to
BC
.
It can be shown that

AX
2
=
2(
2
b
+
2
c
)-
2
a
3
,
d
a
=
S
3a
=
2r
r
a
3(
r
a
-r)
,
d
X
2
=
2
s
+
2
r
+4Rr
6R
.
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 are normalized to a sum of 1.

References

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 9, 2022) faculty.evansville.edu/ck6/encyclopedia.

External Links

Relating Trilinear and Tripolar Coordinates for a Triangle
Triangle Centroid (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Triangle Centroid"​
​http://demonstrations.wolfram.com/BasicParametersOfTheTriangleCentroid/​
​Wolfram Demonstrations Project​
​Published: August 11, 2022