A proof can be given by applying Desargues (see Desargues.html ) on triangles t, t' which are, per construction, point-perspective: J being the intersection point of the lines joining corresponding points {(A, A'), (B, B'), (C, C')}. Hence, by Desargues they are line-perspective: their corresponding side-lines intersect on a line tr(J).

Another proof can be given by considering the basic figure for harmonic conjugate points (see Harmonic.html ). In fact, consider the line of the two intersection points {A*,B*} and show that C* is also on that line. By the referred basic figure points {C',C*} are harmonic conjugate to {A, B}. Hence the bundle of four lines (CA,CB,CC',CC*) is a harmonic one and defines on line A*B* a quadruple of harmonic conjugate points. But line A'B' passes also (basic figure) from the harmonic conjugate of point C' with respect to {A,B}.

A third proof is given in (6) of ProjectiveBase.html .

For this take the intersection points {A*,B*,C*} of L with the sides of the triangle and the corresponding harmonic conjugates {A',B',C'} with respect to point pairs {(B,C),(C,A),(A,B)}. Then lines {AA',BB',CC'} intersect at a point and this is the

[2] The most prominent trilinear polar is perhaps the line at infinity coinciding with the trilinear polar of the centroid of the triangle.

[3] Another interesting case is the symmedian point whose trilinear polar is the Lemoine axis of the triangle.

[4] Also the orthic axis should be mentioned in this respect. It is the trilinear polar of the orthocenter.

[5] It is easily seen that points on the medians of the triangle of reference have trilinear polars which are parallel to the sides to which the medians are drawn. Thus, they pass trhough the same point on the line at infinity determined by the side on which the median. Equivalently, the trilinear poles of lines parallel to one side of the triangle are on the median to that side. This is though an erroneous image. The truth is that the trilinear poles of lines passing through a fixed point are on a conic which passes through the three vertices of the triangle (a circumconic). Here the conic is a degenerate one consisting of two lines: the median and the corresponding side-line on which the median is drawn. For a picture of the general case look at IsotomicConicOfLine.html .

For more general conics resulting as sets of poles of parallel trilinear polars see Trilinear_Polar.html .

In particular, take (A

Thus, taking harmonic associates of J with respect to A'B'C' and repeating the process we get infinite many triangles having the same trilinear polar with respect to the same point. The inverse procedure to create triangles with the same polar is to take the

To variate on this point of view, one can define a

Point J and line tr(J) are called respectively

Harmonic.html

Harmonic_Bundle.html

IsogonalGeneralized.html

IsotomicConicOfLine.html

LineInTrilinears.html

Perspectivity.html

ProjectiveBase.html

TriangleConics.html

Trilinear_Polar.html

TrilinearProjectivity.html

Trilinears.html

Tripole.html

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