[alogo] Triangle Area in Barycentrics

Consider the triangle of reference ABC and a second one DEF, whose vertices have absolute barycentric coordinates (see BarycentricCoordinates.html ) w.r. to ABC: D(dx, dy, dz), E(ex, ey, ez), F(fx, fy, fz). Denote by (Dx, Dy, Dz) etc. the corresponding trilinear coordinates. The basic relations between barycentric and cartesian coordinates are expressed through the vector-relations:
                                                          D = dx*A+dy*B+dz*C,
and analogous formulas for E and F. Writing these equations in matrix form we get:

[0_0]

(See AreaThroughDet.html ) Taking determinants we deduce the formula for the area of DEF, in terms of ABC and its corresponding (absolute) barycentrics:

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Using the basic relation between absolute barycentrics and absolute trilinears:
                                                           ex = (1/2)*a*Ex/area(ABC):

[0_0] [0_1]
[1_0] [1_1]
[2_0] [2_1]

See Also

AreaThroughDet.html
AreaOfPedal.html
BarycentricCoordinates.html
ThreeLines.html

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