[1] Construct as usual the trilinear pole (tripole) of L with respect to the triangle:

- Let {A',B',C'} be the intersection points of L with the side-lines of the triangle.

- Join {A',B',C'} with vertices {A,B,C} respectively to define the side-lines of the

- Lines {AA

[2] Consider an arbitrary point Q on L and the lines {QA

[3] c(Q) passes through Q and is also tangent to L at Q. Thus, it is the unique conic passing through the vertices of ABC and tangent to L at its point Q.

[4] The tripole of L with respect to triangle A''B''C'' is a point Q' on line QP

[5] The tangential triangle A

[6] The locus of perspectors of conics c(Q) is the inscribed conic of the triangle with perspector P

The clue fact is the relation of the triangle ABC to line L, expressed through the tripole P

The particular case of choice is the one for which P

This particular case is studied in AnticomplementaryAndCircumparabola.html . There it is shown that for every point Q on L (i.e. every direction of lines) there is a parabola passing through the vertices of triangles ABC and A''B''C''. Later triangle has as vertices the intersections of lines {QA

All the stated properties are trivially verified in this particular case and transfer to the general case through the projecivity fixing the vertices of ABC and mapping the centroid G onto P

One has only to draw the precevian triangle A

CircumparabolaGeneration.html

TriangleConics.html

TrilinearPolar.html

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