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K1133a :

X(3), X(14), X(15), X(17), X(98), X(3439), X(3480), X(5000), X(5001), X(5981), X(6104), X(16258), X(32628)

K1133b :

X(3), X(13), X(16), X(18), X(98), X(3438), X(3479), X(5000), X(5001), X(5980), X(6105), X(16257), X(32627)

Geometric properties :

K1133a and K1133b are two focal cubics with respective singular foci X(6104) and X(6105), the inverses of X(13) and X(14) in the circumcircle. See Table 55 for other cubics passing through X(5000), X(5001).

They are the isogonal transforms of the two focal cubics K1132a and K1132b.

K1133a and K1133b belong to a pencil of circular cubics which contains K336, K1129, the isogonal transform gK305 of K305, the decomposed cubic (O)∪(E) into the circumcircle and the Euler line. All these circular cubics pass through A, B, C, X(3), X(98), X(5000), X(5001) and have their singular focus on the circle (C) which is the inversive image in (O) of the Fermat axis. See below.

The pencil above is stable under inversion in (O), K336 and (O)∪(E) being the only self-inverse cubics. Any other cubic (K1) has its inverse (K2) in the same pencil and the singular foci F1, F2 – on (C) – lie on a line passing through a fixed point X = X(34218), the inverse of X(15545) in (O). The inverses of F1, F2 in (O) are symmetric about X(115).

Note that the tangents at X(3), X(2079) to (C) concur at X since these two points are the foci of the two self-inverse cubics.

X lies on the line through X(6104), X(6105) since K1133a and K1133b are inverse of one another.

The pencil must also contain the inverse igK305 of gK305, the corresponding foci being X(34010) and X(7575) respectively.


gK305 passes through X(i) for these i : {3, 98, 182, 187, 542, 3455, 5000, 5001, 5939}, singular focus : X(7575).

igK305 passes through X(i) for these i : {3, 6, 67, 98, 183, 524, 2080, 5000, 5001, 16092}, singular focus : X(34010).

(C) passes through X(i) for these i : {3, 187, 1511, 2079, 5961, 6104, 6105, 7575, 12042, 14270, 14702, 14703, 15550, 34010}.

Its center lies on the lines {3, 690}, {23, 9189}, {186, 16230}, {187, 2491}, {511, 14271}, {512, 5926}, {523, 18571}, {669, 9125}, {5466, 11643}, {7492, 9185}, {7771, 14295}, {9033, 12893}, {11616, 25644}.