Here I study the autopolar triangles A'B'C' created as Cevian triangles of points P on a circumconic c

[1] The starting point is a triangle of reference ABC and a point R defining a circumconic c

[2] A''B''C'' denotes the corresponding tangential triangle to c

[3] A'B'C' denotes the Cevian triangle of a point P on c

The following properties related to this configuration are valid:

[4] t

[5] Triangle A''B''C'' is self-polar with respect to c

[6] The perspector is point P' on L and the axis of perspectivity ( Desargues.html ) is the tangent t

[7] The perspectivity H with center P', axis the line t

[8] The same perspectivity maps R to a point R' and triangle ABC to a triangle A

[9] Thus the set of triangles/points/lines {A

[10] Applying the perspectivity repeatedly to triangles ABC, A

[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 t

The proofs of these properties follow from corresponding properties concerning the case in which ABC is equilateral, R is its center, c

In this configuration A''B''C'' coincides with the

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 .

IncircleTangents.html

IsogonalGeneralized.html

ParabolaInscribed.html

Perspectivity.html

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