Basic Parameters of the Symmedian Point

​
classification: even center
standard barycenter:
2
a
2
S
ω

AX
6
 = 1.99187
bc
2(
2
b
+
2
c
)-
2
a
2
S
ω
= 1.99187
d
a
= 1.51807
aS
2
S
ω
= 1.51807
d
X
6
= 3.9239
7.75438S
S
ω
= 3.9239
The angle bisectors of a triangle
ABC
intersect at the incenter
X
1
. The isogonal conjugate
-1
P
of a point
P
is found by reflecting the lines
AP
,
BP
,
CP
about the angle bisectors. The symmedian point
X
6
[1] of
ABC
is the isogonal conjugate of the centroid
X
2
.
Let
d
a
,
d
b
,
d
c
be the exact trilinear coordinates of
X
6
with respect to
ABC
,
d
X
6
=
d
a
+
d
b
+
d
c
,
a
,
b
,
c
be the side lengths opposite the corresponding vertices and let
s
be the semiperimeter of
ABC
,
S=2ABC
,
S
A
,
S
B
,
S
C
be the Conway parameters with
S
θ
=Scotθ
,
ω
be the Brocard angle.
Then, it can be shown that

AX
6
=
bc
2(
2
b
+
2
c
)-
2
a
2
S
ω
,
d
a
=
aS
2
S
ω
,
d
X
6
=
sS
S
ω
.
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 15, 2022) faculty.evansville.edu/ck6/encyclopedia.

External Links

Isogonal Conjugate (Wolfram MathWorld)
Symmedian Point (Wolfram MathWorld)
Brocard Angle (Wolfram MathWorld)
Isogonal and Isotomic Conjugates
Relating Trilinear and Tripolar Coordinates for a Triangle

Permanent Citation

Minh Trinh Xuan
​
​"Basic Parameters of the Symmedian Point"​
​http://demonstrations.wolfram.com/BasicParametersOfTheSymmedianPoint/​
​Wolfram Demonstrations Project​
​Published: August 26, 2022