Here I only notice the fact that the perspector P of the parabola lying on the circumcircle of ABC is also the focus of the parabola.

The fact that P is the focus of the parabola follows from the equality of segments PC'=C'C

The proof follows from Brianchon's theorem and the passing of the circumcircle of A'B'C' through the focus of the parabola (see InconicsTangents.html and ParabolaChords.html ). The theorem of Brianchon implies that lines {FA,FB,FC} pass through the vertices of the tangential triangle. The property of the circumcircle implies that the angles at {A',B',C'} are each π/3 and the triangle is equilateral. Then the Steiner line of F is the directrix and the orthogonal to this line is parallel to the axis of the parabola.

InconicsTangents.html

Parabola.html

ParabolaChords.html

ParabolaInscribed.html

SteinerLine.html

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