Basic Parameters of the Spieker Center

​
classification: odd center
standard barycenter:
2s-a
4s

AX
10
 = 2.71787
1
2
-4
2
a
+4as-3
2
r
-16rR+
2
s
= 2.71787
d
a
= 1.07296
r(2s-a)
2a
= 1.07296
d
X
10
= 4.31815
2
r
-2rR+
2
s
4R
= 4.31815
The Spieker center
X
10
of a triangle
ABC
is the incenter of the medial triangle of
ABC
[1]. It is also the center of the excircles's radical circle.
Let
d
a
,
d
b
,
d
c
be the exact trilinear coordinates of
X
10
with respect to
ABC
and
d
X
10
=
d
a
+
d
b
+
d
c
, let
a
,
b
,
c
be the side lengths opposite the corresponding vertices and let
R
,
r
,
s
be the circumradius, inradius and semiperimeter of
ABC
.
M
A
,
M
B
,
M
C
are the midpoints of
BC
,
CA
,
AB
, respectively.
A
1
B
1
C
1
is the pedal triangle of
X
10
.
Then

AX
10
=
-4
2
a
+4sa+
2
s
-3
2
r
-16Rr
2
d
a
=
r(2s-a)
2a
,
d
X
10
=
2
s
+
2
r
-2Rr
4R
.
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 have 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
Spieker Center (Wolfram MathWorld)
Basic Parameters of the Mittenpunkt

Permanent Citation

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