LineInTrilinears

Line in trilinear coordinates

Given a triangle ABC, draw the line 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. Show that the line passes through three points A*, B*, C* on sides BC, CA, AB respectively. Calculate these points in terms of (U_a,U_b,U_c,U_d) and show that: Varying U_i turns the line about I* (i=A,B,C) and varying U_d produces parallel lines.

A*, B*, C* are the intersection points of the line with the sides of the triangle. They are characterized respectively by x_a =0, x_b =0, x_c = 0. To calculate one of the points, B* say, note that
(1) x_b = 0 => [ U_a*x_a + U_c*x_c = U_d ].
Besides, the point belongs to side AC having equation:
(2) [ a*x_a + c*x_c - 2D = 0 ], where D the area of the triangle.
Thus the intersection point has coordinates determined by the solution of the linear system:
U_a*x_a + U_c*x_c = U_d ,
a*x_a + c*x_c = 2D .
x_a = (c*U_d - 2*D*U_c)/(c*U_a - a*U_c), x_c = (2*D*U_a -a*U_d)/(c*U_a - a*U_c) ,
(x_a/x_c) = (c*U_d - 2*D*U_c)/(2*D*U_a -a*U_d).
Provided (U_a : U_b : U_c) is not a multiple of (a, b, c).
Note that equation a*x_a + b*x_b + c*x_c - 2D = 0 is valid for the trilinears of every point of the plane.
The line through B, whose points have distance ratios from the sides BA, BC given by (x_a/x_c) intersect side AC at the point P=B*, dividing AC in ratio
r_b = (B*A/B*C) = -(x_c/sinA)/(x_a/sinC) = -(x_c/x_a)*(sinC/sinA) = -(x_c/x_a)*(c/a) =
= -((2*D*U_a -a*U_d)/(c*U_d - 2*D*U_c))*(c/a) = (h_a*U_a - U_d)/(h_c*U_c-U_d), where h_a is the height to side a.

The intersection point B* of line CA and the line U_a*x_a + U_b*x_b + U_c*x_c = U_d is determined on line CA through the ratio r_b. turning property. The other points C*, A* are determined by expressions resulting, from the previous by cyclic substitution of the letters a,b,c. But the equation for r_b is independent of U_b This proves the statement about the
Note that there is a double infinity of values of U_i's defining the same line. In fact, the line is uniquely defined by the (oriented) ratios r_b = B*A/B*C, r_c = C*B/C*A, r_a = A*C/A*B, satisfying also the Menelaus condition r_a*r_b*r_c = 1. It follows that for fixed r_i's and arbitrary y, and non zero x, all the values of the coefficients given by the following equations determine the same line:
U_d = y,
U_a = (x+y)/h_a,
U_b = (x*r_c + y)/h_b,
U_c = (x*r_b*r_c + y)/h_c.

References

Yiu, P. Euclidean Geometry Notes (pdf), chapter 8, p. 104. http://www.math.fau.edu/yiu/EuclideanGeometryNotes.pdf

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Line in trilinear coordinates

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References