Relating Trilinear and Tripolar Coordinates for a Triangle

​
d
a
d
b
d
c
from definition
-1.8401
2.91043
4.73749
from formulas
-1.8401
2.91043
4.73749
Given a triangle
ABC
, the trilinear coordinates of a point
P
are the signed distances to the extended sides. Denote the signed distances of
P
to
BC
,
CA
and
AB
by
d
a
,
d
b
and
d
c
, respectively. If
P
and the incenter
I
are in the same half-plane determined by a side, the signed distance to that side is positive; otherwise, it is negative.
The tripolar coordinates of the point
P
are its distances to the vertices of the triangle, given by
s
A
=PA
,
s
B
=PB
and
s
C
=PC
.
The Conway triangle notation relates the sides to twice the area of the triangle, denoted by
S
:
S
A
=
2
b
+
2
c
-
2
a
2
,
S
B
=
2
c
+
2
a
-
2
b
2
,
S
C
=
2
a
+
2
b
-
2
c
2
,
S=
S
A
S
B
+
S
B
S
C
+
S
C
S
A
.
These definitions imply the following formulas between the trilinear and tripolar coordinates:
d
a
=
2
a
(
S
A
-
2
AP
)+
S
B
2
CP
+
S
C
2
BP
2aS
,
d
b
=
2
b
(
S
B
-
2
BP
)+
S
C
2
AP
+
S
A
2
CP
2bS
,
d
c
=
2
c

S
C
-
2
CP
+
S
B
2
AP
+
S
A
2
BP
2cS
.

External Links

Circumcircle (Wolfram MathWorld)
Incircle (Wolfram MathWorld)
Mapping a 3D Point to a 2D Triangle
Trilinear Coordinates (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan
​
​"Relating Trilinear and Tripolar Coordinates for a Triangle"​
​http://demonstrations.wolfram.com/RelatingTrilinearAndTripolarCoordinatesForATriangle/​
​Wolfram Demonstrations Project​
​Published: April 26, 2022