Barycentric Coordinates
Barycentric coordinates are triples of numbers 
corresponding
to masses placed at the vertices of a reference triangle 
.
These masses then determine a point 
, which is the geometric centroid of the three masses and is
identified with coordinates 
. The
vertices of the triangle are given by 
, 
, and 
. Barycentric coordinates were
discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993).
To find the barycentric coordinates for an arbitrary point 
, find 
and 
from the point
at the intersection of the line 
with the side 
, and then
determine 
as the mass at 
that will balance
a mass 
at 
, thus making 
the centroid (left figure). Furthermore,
the areas of the triangles 
, 
, and 
are
proportional to the barycentric coordinates 
, 
, and 
of 
(right figure;
Coxeter 1969, p. 217).
Barycentric coordinates are homogeneous, so
 |
(1)
|
for 
.
Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous barycentric coordinates. Barycentric coordinates normalized so that
 |
(2)
|
so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates (Coxeter 1969, p. 218). Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem (Coxeter 1969, pp. 219-221).
(Not necessarily homogeneous) barycentric coordinates for a number of common centers are summarized in the following table. In the table, 
, 
, and 
are the side lengths
of the triangle and 
is its semiperimeter.
| triangle center | barycentric coordinates |
circumcenter  | ( ,
 ,  ) |
excenter  |  |
excenter  |  |
excenter  |  |
| Gergonne point Ge | ( ,  ,  ) |
incenter  |  |
| Nagel point Na |  |
orthocenter  | ( ,  ,
 ) |
symmedian point  |  |
triangle centroid  |  |
In barycentric coordinates, a line has a linear homogeneous equation. In particular, the line joining points 
and
has equation
 |
(3)
|
(Loney 1962, pp. 39 and 57; Coxeter 1969, p. 219; Bottema 1982). If the vertices 
of a triangle 
have
barycentric coordinates 
, then
the area of the triangle is
 |
(4)
|
(Bottema 1982, Yiu 2000).
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