Tripole

Given a triangle ABC, draw the line L with given equation U_a*x_a + U_b*x_b + U_c*x_c = U_d in trilinear coordinates. The trilinear coordinates x_a, x_b, x_c denote the oriented distances from the sides of the triangle.This line passes through three points A*, B*, C* on sides BC, CA, AB respectively (see LineInTrilinears.html ). (A*, B*, C*) are correspondingly the harmonic conjugates of points (A',B',C') with respect to the pairs of points (B,C), (C,A), (A,B). Lines AA', BB', CC' intersect at a point M with trilinear coordinates (x : y : z). Below is given the calculation of the trilinears of M in terms of the coefficients U_i. M is the tripole of the line L through A*,B*,C*. L is in turn the [trilinear polar] of M with respect to the trianlge ABC.

The calculation of the trilinears of M (see aforementioned reference) gives:
(x : y : z ) = ( 1/(a*(h_a*U_a-U_d)) : 1/(b*(h_b*U_b-U_d)) : 1/(c*(h_c*U_c-U_d)) ).
The relations can be solved w.r. to the U_i's expressing them as functions of the trilinears of M:
U_d = k ,
U_a = (m/(a*x) + k)/h_a,
U_b = (m/(b*y) + k)/h_b,
U_c = (m/(c*z) + k)/h_c,
where k and the non zero m are arbitrary. In particular, for k = 0, the equation takes the form:
U_a*x_a + U_b*x_b + U_c*x_c = 0, where (U_a : U_b : U_c) = (1/(a*x) : 1/(b*y) : 1/(c*z)).

References

Yiu, P. GeometryNotes020402 (pdf). http://www.math.fau.edu/yiu/GeometryNotes020402.pdf

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Tripole

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References