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X(3), X(20), X(67271)

infinite points of K003

vertices of the CircumNormal triangle N1N2N3

other points below

Geometric properties :

Q1, Q2, Q3 are the points on the circumcircle whose Simson lines S(Q1), S(Q2), S(Q3) concur at O.

Q1, Q2, Q3 lie on the rectangular hyperbolas homothetic to the Jerabek and Kiepert hyperbolas passing through {3, 20, 74, 2574, 2575} and {2, 20, 98, 3413, 3414} respectively. The first hyperbola is (H) on the figure and it is the image of the Jerabek hyperbola in the translation that sends H onto O.

Note that the Kiepert parabola is inscribed in this triangle Q1Q2Q3 (and also in ABC).

Q1, Q2, Q3 also lie on nK0(X6, X7735) and on nK0+(X3, X2). These two cubics meet at six points on (O) and three other points on the line X(6), X(523).

The antipodes of Q1, Q2, Q3 on (O) lie on the Orthocubic K006.

K1386 is the McCay cubic of this triangle Q1Q2Q3 whose orthocenter is X(20). Its sidelines are parallel to the asymptotes of the Orthocubic K006.

It follows that K1386 passes through the in/excenters J, Ja, Jb, Jc of Q1Q2Q3 which lie on the Stammler hyperbola and also on (H') passing through X(20), X(3164), with center X(925). Note that the Stammler hyperbola is the polar conic of O in both cubics K003 and K1386, hence the two cubics have a triple contact at O.

The vertices R1, R2, R3 of the cevian triangle of O with respect to Q1Q2Q3 are obviously on the cubic.

See the analogous K1387.