[alogo] 1. Autopolar or self-polar triangles

A triangle is autopolar or self-polar with respect to a conic if each side-line of the triangle is the polar of the opposite lying vertex.

Here I study the autopolar triangles A'B'C' created as Cevian triangles of points P on a circumconic cR of a triangle of reference ABC. These triangles generate a whole sequence of conics and other self-polar triangles. Any two triangles of this sequence are perspective and the conics are all tangent to cR at its point P.

[1] The starting point is a triangle of reference ABC and a point R defining a circumconic cR with this point as perspector.
[2] A''B''C'' denotes the corresponding tangential triangle to cR with perspector R. L=tr(R) denotes the trilinear polar of R with respect to ABC which is the same with the trilinear polar tr''(R) of R with respect to A''B''C''.
[3] A'B'C' denotes the Cevian triangle of a point P on cR with respect to ABC. tp denotes the tangent to cR at P.
The following properties related to this configuration are valid:

[4] tp is the trilinear polar tr''(P') with respect to A''B''C'' of a point P' on L. It is also the trilinear polar tr'(P') of the same point P' with respect to A'B'C'.
[5] Triangle A''B''C'' is self-polar with respect to cR, perspective to it and its side-lines pass through the vertices of ABC.
[6] The perspector is point P' on L and the axis of perspectivity ( Desargues.html ) is the tangent tP.
[7] The perspectivity H with center P', axis the line tP and homology coefficient -0.5 (see Perspectivity.html ) maps triangle A''B''C'' to A'B'C'.
[8] The same perspectivity maps R to a point R' and triangle ABC to a triangle A0B0C0. The trilinear polar of R' with respect to both triangles A'B'C' and A0B0C0 is again line L.
[9] Thus the set of triangles/points/lines {A0B0C0, A'B'C', P, tP, P', L} has elements related through the same relations as the set {ABC, A''B''C'', P, tP, P', L}. The circumconic cR' of A0B0C0 with perspector R' is the image of cR under the perspectivity H. Conics cR and cR' are tangent at P.
[10] Applying the perspectivity repeatedly to triangles ABC, A0B0C0=H(ABC), ... we obtain a sequence of triangles all perspective to ABC and a sequence of circumconics cR, cR'=H(cR), ... all tangent to tP at P.
[11] Analogously applying the perspectivity H to triangles A''B''C'', A'B'C'=H(A''B''C''), ... we obtain a sequence of triangles all perspective to A''B''C'', all passing through two points of line tP and with perspectors R, R'=H(R), ... .

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]
[3_0] [3_1] [3_2] [3_3]

The proofs of these properties follow from corresponding properties concerning the case in which ABC is equilateral, R is its center, cR is its circumcircle and P a point on this circumcircle. To reduce to that case simply use the well defined projectivity F which maps the vertices of ABC to the vertices of the equilateral and maps also point R to the center of the equilateral. Then line L=tr(R) maps to the line at infinity and the configuration translates to the one studied in IncircleTangents.html . The properties here translate to the properties there, studied in section-2, and the proofs follow from their inherent projective invariance.

[alogo] 2. The case of Steiner

Apply the previous discussion to the particular configuration resulting from a triangle ABC, its outer Steiner ellipse (c) and a point P on it. In this case the perspector R of the initial conic coincides with the centroid G of the triangle. Point P' is a point at infinity and the perspectivity H becomes an affinity. H is characterized by leaving tP pointwise fixed and mapping every point Q to a point Q'=H(Q) such that QQ' is in the direction defining the point at infinity P' and QQ' is divided by tP at a point Q0 such that QQ0/Q0Q'=2.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

In this configuration A''B''C'' coincides with the antiparallel triangle of ABC and the lines joining corresponding vertices of triangles A''B''C'', A'B'C'=H(A''B''C''), ... are parallel in the direction defining the point at infinity P'. The inconics of these triangles are the corresponding inner Steiner ellipses. In this case also P is the isotomic conjugate of P' with respect to any triangle of the sequence defined above (see section-10 of IsogonalGeneralized.html ).
The previous figure is related to properties of parabolas inscribed in triangle ABC. All these parabolas have perspectors P lying, as above, on the outer Steiner ellipse of ABC. This subject is discussed in ParabolaInscribed.html .

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