A = (a, d)
B = (b, e)
C = (c, f)
A(triangle) = length of cevian * width of triangle with respect to cevian - that is, the length of the projection of the opposite side onto the perpendicular to the cevian
BC: y = ((f-e)/(c-b))(x-b)+e
= ((f-e)/(c-b))x - b((f-e)/(c-b)) + e
= ((f-e)/(c-b))x - ((bf - be)/(c-b)) + e
= ((f-e)/(c-b))x - ((bf - be + be - ec)/(c-b))
Length of vertical cevian: d - ((f-e)/(c-b))a + ((bf - ec)/(c-b))
Width of triangle = c - b
Area = d(c-b) - a(f-e) + bf - ec
= d(c-b) + e(a-c) + f(b-a)
=
|a d 1|
|b e 1|
|c f 1|