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X(3), X(4), X(20), X(1158), X(6261), X(9732), X(9733), X(61092), X(61093), X(61094), X(61095), X(61096), X(61097)

infinite points of the altitudes of ABC

points T1, T2, T3 (apart A, B, C) of K006 on (O) and their antipodes S1, S2, S3

other points below

The points T1, T2, T3 lie on the rectangular hyperbola H(T) through {3, 4, 110, 155, 1351, 1352, 2574, 2575}, center X(113). Recall that K006 passes through the midpoints of T1T2T3.

The points S1, S2, S3 lie on the rectangular hyperbola H(S) through {3, 20, 74, 2574, 2575, 12163, 33878, 46254}, center X(16111).

H(T) and H(S) are obviously symmetric about O and homothetic to the Jerabek hyperbola.

Geometric properties :

The isogonal conjugation with respect to the triangle (T) = T1T2T3 transforms K006 into K1362. See CL076 for a generalization and other analogous cubics.

Recall that the conic inscribed in ABC and (T) is the MacBeath inconic with center X(5), perspector X(264), foci O and H.

K1362 is a central cubic with center O and inflexional tangent passing through X(64), X(154).

The asymptotes of K1362 are those of the Darboux cubic K004 and the remaining common points of the two cubics are X(3), X(4), X(20).

K1362 is a member of the pencil of cubics generated by K004 and the union of the line at infinity (twice) with the Euler line. This pencil also contains K1361.

K1362 meets (O) at T1, T2, T3 on K006 and their antipodes S1, S2, S3. The tangents at X(6261), T1, T2, T3 concur at X(11472) and the tangents at X(1158), S1, S2, S3 concur at X(35237).

Other points on K1362

Q1 on the lines {3, 66}, {4, 372}, {20, 488}, {30, 591}, {76, 490}, {98, 485}, {99, 489}, SEARCH = -13.3691252010417.

Q2 on the lines {3, 66}, {4, 371}, {20, 487}, {30, 1991}, {76, 489}, {98, 486}, {99, 490}, SEARCH = 26.9338509894344.

These two points are symmetric about O. They are the (T)-isogonal conjugates of X(371) and X(372) respectively. They are X(61097) and X(61096) in ETC.

***

K1362 meets K006 at X(3), X(4), T1, T2, T3 and four other points on the parallels at O to the asymptotes of the Jerabek hyperbola and on the rectangular hyperbolas which are their isogonal transforms in (T). These points are not all real. See R1, R2 on the figure. They also lie on the common axes of the inconics with center O in ABC and (T).

K006 and K1362 generate a pencil of cubics passing through these same nine points. This pencil is stable under isogonal conjugation in (T). It contains two invariant cubics, namely :

• a focal cubic (F) with singular focus X(110) which is K187 with respect to (T).

• a stelloid (S) which is K003 with respect to (T). This passes through the infinite points of K003 and the vertices of the CircumTangential triangle.

See Table 54, line Q = X(3) and Table 81 where an analogous configuration is described.