By

[2] The trilinears are defined modulo a non-zero multiplicative constant d. Thus (d*x, d*y, d*z) are considered to define the same point P with (x,y,z).

[3] There are also the

[4] To determine the absolute trilinears from an arbitrary trippel (x,y,z) = d*(x

[5] Trilinears are the

[6] Points on the sides of ABC have corresponding coordinate zero. For example, P on BC is characterized by trilinears of the form (0, y, z). These are directly related to the oriented ratio PB/PC = - (z/sinB)/(y/sinC) = -(z/y)*(sinC/sinB) = -(z/y)*(c/b) => z/y = -(PB/PC)*(b/c).

[7] Drawing a parallel to BC from P' we see that all points P' of line AP are characterized by the equation z/y = - k <==> z + k*y = 0, with k = (PB/PC)*(b/c). Thus, the

[8] In general a

[9] The coefficients {u,v,w} are directly related to the coordinates of the intersection points of L with the sides of the triangle of reference: A'(0,y

[10] The two last equations are equivalent to

The interpretation of the coordinates in terms of a model of the projective plane can be described as follows:

[11] In the three dimensional space R

[12] Points [X] = [x,y,z] with a*x+b*y+c*z = 0 represent the

[13] Each ordinary line L: u*x+v*y+w*z = 0 of (E) intersects the line at infinity a*x+b*y+c*z = 0 at the point (at infinity), whose coordinates are given by the vector product U

[14] This point represents the

[15] The system is a compatible one of rank two, thus defining a one-parameter family of lines. The U's representing the solutions define a vector orthogonal to S, thus a linear combination of the form U = rF + t(F

[16] For a particular point P

[17] In absolute trilinears the inner product (F,X) = 2*D, hence for these coordinates the equation of the line (U,X) = 0 becomes (F

[18] The trilinears, as with any projective coordinate system, allow to write the points of the plane as linear combinations P = xA+yB+zC. This is handy when calculating cross ratios along lines, as for example with the points {Q,R,S} called

[19] In the same spirit we can write P = Q+xA and for the harmonic conjugate of P w.r. to A,Q: P

[20] The above picture (to be encountered in many places, see for example in TriangleConics.html ), which

[21] It is also easily seen that line L

[22] The selection of the incenter as the central point (or

[23] Selecting two such coordinate systems {A,B,C,D} and {A',B',C',D'} where in each system D (resp. D') plays the role of coordinator (i.e. D = A+B+C, resp. D' = A'+B'+C'), the corresponding coordinates are related by a matrix relation of the form X' = MX, the matrix M being invertible and having columns the coordinates of {A,B,C} with respect to {A',B',C'}. In particular, systems with A=A', B=B', C=C' but different D, D' have a diagonal matrix, whose entries are the coordinates of D with respect to {A',B',C'}.

[24] For example, the relation between Barycentric (X=(x,y,z)) and standard trilinears (X'=(x',y',z')), as defined in [1], is

X' = (x/a, y/b, z/c), since (1/a, 1/b, 1/c) are the trilinears of the centroid.

More general, selecting coordinator D whose trilinears are (d

A line with equation px+qy+rz=0, has in trilinears the equation (p/d

A conic with equation pyz+qzx+rxy=0, has in trilinears the equation (pd

[25] In particular, the line at infinity is easily calculated in barycentrics to have the equation x+y+z=0.

It follows, that in trilinears it has the equation ax'+by'+cz'=0, and in more general systems (see IsogonalGeneralized.html ) with coordinator D (=(d

The circle, which in the system with coordinator D equal to the symmedian point has equation yz+zx+xy=0, in trilinears has the representation ay'z'+bz'x'+cx'y'=0 and a

BarycentricsFormulas.html

Harmonic.html

IsogonalGeneralized.html

ProjectiveBase.html

ProjectiveCoordinates.html

TriangleConics.html

TrilinearPolar.html

TrilinearsRelatedToOthers.html

Tripole.html

Yiu, P.

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