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X(1), X(4), X(40), X(516), X(972), X(1155), X(1156), X(3160)
X1-OAP points, see also Table 53
S1 = X(15725) described below
A X1-OrthoAntiPedal point P is a point whose antipedal triangle has orthocenter the incenter X1 of ABC.
There are four such points : X1 itself (this is obvious) and three other points P1, P2, P3.
These three points lie :
• on the circle (C) with center X(3), radius R + r, passing through the point S2 on the lines X(1)X(528), X(3)X(8), X(11)X(214), etc, SEARCH = 1.64900952318758. S2 is now X(10609) in ETC (2016-10-26).
• on the rectangular hyperbola (H) homothetic to the Feuerbach hyperbola and passing through X(1), X(8), X(20), X(72), X(224), X(442), X(1490), X(2582), X(2583), etc, and S2 as well.
In other words, P1, P2, P3 are the common points (apart S2) of (C) and (H).
See Table 53 for other curves passing through these points.
K806 is a circular cubic meeting (C) at four finite points namely P1, P2, P3 and S1 = X(972)X(3160) /\ X(4)X(1156), etc, now X(15725) in ETC.
Its singular focus is F = X(1)X(41) /\ X(214)X(3126), SEARCH = 1.50540762290438, now X(11712) in ETC.
There are actually infinitely many circular circum-cubics of a same pencil passing through X(1), P1, P2, P3 and K806 is the only one which is a K0.
The pencil contains three psKs (each passing through one antipode of A, B, C on the circumcircle) and three (very complicated) nKs.
Their singular foci lie on the circle with center the midpoint X(1385) of X(1)X(3) and radius R/2 which passes through X(214) and obviously F. The cubic with focus X(214) contains X(3), X(102), X(517).
See also K825.
The Pelletier strophoid K040 and K806 generate a pencil of circular circum-cubics that contains the three pKs K949, K950, K951. The singular focus of each cubic of the pencil lies on the circle passing through X(105), X(1001), X(11712), X(14074) which is orthogonal to the circumcircle.
This pencil is stable under isogonal conjugation and the self-isogonal cubics are K040 and K951 each with singular focus on the circumcircle, namely X(105), X(14074) respectively. More generally, the singular foci of a cubic and its isogonal transform are inverse in the circumcircle.