## An Example of Barycentric Coordinates and Triangle Area

The 3 sides of triangle ABC are each divided in a ratio of 2 to 1. In other
words, on side AB, AC'/AB = 1/3, or AC'/C'B = 1/2. This enables one to compute
the barycentric coordinates x, y, z of any of the points E, F, G. In turn, it
is possible to compute the ratios of the areas of subtriangles to the area of
ABC.

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#### Computing the Barycentric Coordinates of F from the ratios on two
sides

F is the intersection of BB' and CC'. Thus we have 1/2 = AC'/C'B
= y/x and 1/2 = CB'/B'A = x/z. Thus x = 2y and also z = 2x. Thus if y = 1 (an
arbitrary choice), then x = 2 and z = 4. Since the sum x+y+z i supposed to be
1, we divide by 7 = 1+2+4 to get the true value x = 2/7, y = 1/7, z = 4/7.

This means that the areas of the 3 triangles divided by area ABC
are equal to the same numbers 2/7, 1/7, 4/7 as shown in the figure, no matter
where A, B and C are dragged.

**Note**: By Ceva, if the line AF intersects BC at Q, then BQ/QC
= z/y = 4/1 = 4.

#### Computing the Barycentric Coordinates of E and F and the area of
DEF

The same computations give barycentric coordinates for E and G (same
numbers, different order). For example, the barycentric coordinates of D are
(x,y,z) = (4/7,2/7,1/7).

Taking the area of ABC to be 1, the sum of the areas of triangles
DAB + EBC + FCA = 3(2/7) = 6/7. Thus the area of DEF = 1/7.

James King, 1/15/2003

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