Consider an arbitrary triangle ABC and a circumconic k of it with perspector at the point S. Let s be the trilinear polar of S with respect to ABC. Points {A*,B*,C*} are respectively the intersections of {AS,BS,CS} with s, and points {A0,B0,C0} are the other intersection points of the same lines with the circumconic k. Fix a vertex, A say and its opposite side BC. The following results are valid also for corresponding constructions relative to the other sides. [1] The conic kA tangent to lines {AB,AC} at {B,C} and passing through A* is tangent there to the trilinear polar s. [2] Le point P vary on line BC and P' be the intersection of A*P with the polar s' of S with respect to kA. Let also Q be the other intersection point of line A0P with the conic k. The diagonals of quadrangle SPQP' intersect at a point W lying on the conic kA. Note that the polar s' is also the harmonic conjugate of the trilinear polar s with respect to the pair of lines A'A and A'S. [3] Consider the other than A* intersection point A2 of line AS with the conic kA, then the cross ratio (A,A2,S,A*)=4. [4] Let A1 be the intersection of line AS with the polar s' of S with respect to kA. The projectivity FA is defined by the conditions (i) it fixes points {B,C,S}, and (ii) it maps A* to A1. FA is a perspectivity mapping the circumconic k to kA. [5] The three analogously defined conics {kA, kB, kC} are permuted by the dihedral group generated by the three projective reflexions {RA, RB, RC}. RA is defined as the projectivity fixing {A,S} and interchanging {B,C}. RA coincides with the harmonic perspectivity with center at A' and axis the line AS, which is common polar of A' with respect to both conics k and kA. Analogously are defined the other two reflexions.
The proofs of the statements follow immediately by applying an appropriate projectivity to the corresponding configuration resulting when S obtains the position of the centroid of ABC. In that case the three conics {kA,kB,kC} become the three Artzt parabolas of the triangle a and the corresponding discussion can be found in ArtztSteiner.html .
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Three conics associated to a circumconic ![[0_0]](ArtztSteiner2_files/ArtztSteiner2_0_0_0.png)
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![[1_0]](ArtztSteiner2_files/ArtztSteiner2_0_1_0.png)
![[1_1]](ArtztSteiner2_files/ArtztSteiner2_0_1_1.png)
![[1_2]](ArtztSteiner2_files/ArtztSteiner2_0_1_2.png)