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X(3233), X(3234) foci of the Kiepert and Yff parabolas

vertices of the cevian triangle of X(69)

other points below

Q078 is the locus of vertices of inscribed parabolas.

It is known that a parabola inscribed in ABC must have its focus F on the circumcircle and its perspector P on the Steiner ellipse. Its directrix D is the Steiner line of F and its vertex V is the midpoint of F and the foot on D of the perpendicular at F to D (the axis of the parabola).

Q078 is a bicircular quartic that meets the circumcircle at the same points as nK0(X6, X15302) with X(15302) on the lines X(2)X(39), X(23)X574), etc.

In ETC, we find a list of pairs (F, V) = (X(i), X(j)) for these (i, j) :

(74, 12079), (98, 15630), (99, 15631), (100, 15632), (101, 3234), (102, 15633), (103, 15634), (104, 15635), (105, 15636), (106, 15637), (110, 3233), (111, 15638), (112, 15639).