Homogeneous barycentric coordinates are barycentric coordinates normalized such that they become the actual areas of the subtriangles. Barycentric coordinates normalized so that
 |
(1) |
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).
The homogeneous barycentric coordinates corresponding to exact trilinear coordinates 
are 
, where
The homogeneous barycentric coordinates for some common triangle centers are summarized in the following table, where 
is the circumradius of the
reference triangle.
See also
Areal Coordinates, Barycentric Coordinates, Trilinear Coordinates
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References
Coxeter, H. S. M. "Barycentric Coordinates." ยง13.7 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 216-221, 1969.
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Homogeneous Barycentric Coordinates
Cite this as:
Weisstein, Eric W. "Homogeneous Barycentric Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HomogeneousBarycentricCoordinates.html