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X(1), X(2), X(7), X(57), X(145), X(174), X(1488), X(2089), X(19604), X(44301)

infinite points of the Thomson cubic.

vertices of the intouch triangle

points on (O) and pK(X6, X3873)

points on the Steiner ellipse and pK(X2, X3875)

K365 is a member of the classes CL040 and CL042.

It has three real asymptotes parallel to those of the Thomson cubic K002.

The isogonal transform of K365 is K761 = pK(X41, X1) and its isotomic transform is pK(X312, X75).

The symbolic substitution SS{a -> a^2} maps K365 onto K233.

All the cubics in {K201, K365, K747, K748, K761, K1077, K1078, K1079, K1082} are anharmonically equivalent.

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K761 is the locus of the pseudo-poles of central cubics psK with center X(1).

The loci of the pseudo-pivots and the pseudo-isopivots are K365 and pK(X220, X1) respectively.

See Pseudo-Pivotal Cubics and Poristic Triangles and Central cubics.

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Locus property (Angel Montesdeoca, 2022-04-15)

Denote by DEF the cevian triangle of P. IC cuts the perpendicular to IB though D at Ab, and IB cuts the perpendicular to IC though D at Ac. Points Bc, Ba and Ca, Cb are defined cyclically. The triangle formed with lines AbAc, BcBa, CaCb is perspective to ABC if and only if P lies on K365.

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A remarkable pencil of cubics

K365 belongs to the pencil of circum-cubics generated by K002 and the decomposed cubic which is the union of the line at infinity and the circum-conic with perspector X(513). This conic passes through X(i) for these i : 1, 2, 28, 57, 81, 88, 89, 105, 274, 277, 278, 279, 291, 330, 367, 955, 957, 959, 961, 985, 1002, 1022, 1123, 1170, 1219, 1224, 1255, 1257, 1258, 1280, 1336, 1390, 1422, 1432, 1929, 2006, 2224, 2282, 2306, 2362, 2401, 2982, 2990, 3227, etc.

The base points are A, B, C, X(1), X(2), X(57) and the infinite points of K002. This pencil of K0 cubics contains :

• three pKs namely K002, K308, K365.

• three nK0s namely the decomposed cubic above and (K1) = nK0(X1381, R1 = X1381 ÷ X1382), (K2) = nK0(X1382, R2 = X1382 ÷ X1381). Recall that X(1381) and X(1382) lie on the line X(1)X(3) and on the circumcircle (O). Their isogonal conjugates X(3307), X(3308) are the infinite points of the Feuerbach hyperbola.

• one K+ with asymptotes concuring at X(4383), a point on {1,210}, {2,6}, {57,122) and many other lines.

Properties of R1, R2

• R1, R2 are the barycentric quotients of two antipodes on (O) and they are obviously isotomic conjugates.

• R1, R2 lie on the line {190,644} and on the lines {390,3308}, {390,3307} respectively.

• R1, R2 lie on the circum-conic that is the isotomic transform on the line above. Its perspector is X(11) and its center is X(650).

• R1, R2 are the trilinear poles of the Simson lines (S1), (S2) of X(1381), X(1382) respectively. Hence they lie on the Simson cubic K010.

• R1, R2 lie on every isotomic pK with pivot on the line {190,644}. The most interesting is pK(X2, X883) passing through X(2), X(7), X(8), X(883), X(885), X(2398), X(2400).

Properties of (K1) and (K2)

(K1) passes through the base points above and X(1381).

It contains the traces A1, B1, C1 of (S1) and the points D1, E1 on (O) and on the parallel (L1) at G to (S1).

P1, P2 are the third points on the lines {1,2}, {2,57} respectively and X(100), X(1381), P1, P2 are collinear.

P1, P2 also lie on the lines passing through X(518) and X(2446), X(2447) respectively.

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(K2) passes through the base points above and X(1382).

It contains the traces A2, B2, C2 of (S2) and the points D2, E2 on (O) and on the parallel (L2) at G to (S2).

P3, P4 are the third points on the lines {1,2}, {2,57} respectively and X(100), X(1382), P3, P4 are collinear.

P3, P4 also lie on the lines passing through X(518) and X(2447), X(2446) respectively.

Note that the barycentric product P1 x P3 is X(644) and P2 x P4 is X(651).

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Generalization

The points X(1381), X(1382) can easily be replaced by two antipodes O1, O2 on (O) and most of the properties above are generalized. Note that the barycentric product O1 x O2 lies on the circum-conic with perspector X(184).

Let R1 = O1 ÷ O2 and R2 = O2 ÷ O1, two isotomic conjugates on K010, which are the roots of the cubics (K1) = nK0(O1, R1) and (K2) = nK0(O2, R2) and the trilinear poles of the Simson lines of O1 and O2.

(K1) and (K2) generate a pencil of circum-cubics that contains K002 and two other (not always real) pKs with poles Ω1, Ω2 on K002 and collinear with O, with pivots P1, P2 on K007 and collinear with X(69) hence isotomic conjugates. The barycentric product Ω1 x Ω2 lies on K009.

This pencil contains a third nK0 which is the union of the line at infinity and a circum-conic passing through G. This is the circum-conic whose perspector is the infinite point of the perpendicular bisector of O1O2.

It also contains a K+ with asymptotes concurring on the line GK.