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X(1), X(41), X(56), X(208), X(603), X(16944), X(16945), X(17103), X(17104), X(17105), X(17106), X(17107), X(17108), X(17109), X(17110)

Q1 = a^3 (a+b-2 c) (a-2 b+c) (a^2-b^2+b c-c^2) : : = X(16944)

Q2 = a^3 (a+b-3 c) (a+b-c) (a-3 b+c) (a-b+c) : : = X(16945)

vertices of the cevian triangle of X(57)

points of pK(X604, X56) on (O), see below

Geometric properties :

A variable line (L) passing through the incenter X(1) of ABC meets the Thomson cubic K002 again at M1, M2 whose barycentric product is S.

When (L) rotates around X(1), the locus of S is K967, a nodal cubic with node X(56).

The pseudo-isopivot is X(31) hence its polar conic (C) is the circum-conic passing through X(56) meeting K967 again at Q1 as above.

K967 is the barycentric product of :

• X(1) and K577 = psK(X56, X7, X2),

• X(57) and K259 = psK(X55, X2, X1),

• X(6) and K970 = psK(X7, X85, X7), a nodal cubic with node X(7), with asymptotes concurring at the midpoint X(6173) of X(2)X(7), passing through X(7), X(9), X(75), X(77), X(342),

• X(7) and psK(X2175, X1, X6), a nodal cubic with node X(6), passing through X(6), X(9), X(48), X(1253), X(1419), X(2191), X(2331).

K967 is also the X(31)-isoconjugate of K220 = psK(X55, X2, X6), the X(604)-isoconjugate of K360 = psK(X56, X7, X1).

The isogonal transform of K967 is K971 = psK(X8, X75, X1), a nodal cubic with node X(8), with asymptotes concurring at the midpoint X(3679) of X(2)X(8), passing through X(1), X(8), X(85), X(271), X(318).

***

Points on the circumcircle (O) and related cubics

A cubic pK(Ω, P) meeting (O) at the same points as K967 must have :

• its pole Ω on psK(X32 x X1408, X1408, X604) passing through X(604), X(1333),

• its pivot P on psK(X1408, X1014, X1) passing through X(1), X(56), X(58).

The more simple is probably pK(X604, X56) passing through X(1), X(56), X(266), X(978).

Also :

• pk1 = pK(Ω1 = a^3 (a^2+a b+a c-2 b c) : : , X1), Ω1 = X(16946) now in ETC,

• pk2 = pK(Ω2 = a^4 (a+b) (a+b-c) (a+c) (a-b+c) : : , X58), Ω2 = X(16947) now in ETC,

• pk3 = pK(X1333, P3 = a (a+b) (3 a-b-c) (a+c) : : ), P3 = X(16948) now in ETC,

with rather simple poles or pivots.

***

Consider the net of cubics psK(Ω, X7, X1) with pseudo-pole Ω. See § 4.2.3 in Pseudo-Pivotal Cubics and Poristic Triangles.

The polar conic of X(1) in this psK is degenerated into two lines if and only if Ω lies on K967. One of these lines (L1) passes through X(1) and is the tangent to the psK at this point. The other line (L2) usually does not contain X(1) which turns out to be then a point of inflexion on the psK except when Ω = X(56) in which case it is a node on the cubic. This is K360 = psK(X56, X7, X1). When Ω traverses K967, the envelope of (L2) is the parabola with focus X(1054) and directrix the line X(1)X(2).

***

Generalization

A variable line (L) passing through the incenter X(1) of ABC meets the cubic pK(X6, Q) again at M1, M2 whose barycentric product is S. Denote by M the Q-Ceva conjugate of X(1), a point on the pK.

When (L) rotates around X(1), the locus of S is the nodal psK with

• pseudo-pole Ω = X(560) ÷ M (barycentric quotient),

• pseudo-pivot P = Q x M* (barycentric product of Q and isogonal conjugate M* of M),

• pseudo-isopivot X(1) x Q*,

• node X(1) x M*.

The isogonal transform of this psK is a psK+ with

• pseudo-pole Ω1 = M ÷ X(1) = anticomplement of (X(1) ÷ Q),

• pseudo-pivot P1 = Q ÷ X(1),

• pseudo-isopivot M ÷ Q,

• node N = Ω1,

• asymptotes concurring at X, the midpoint of X(2), N.

Now, if the node N is given, this psK+ is psK(N, tcN, N) corresponding to pK(X6, X(1) x tcN).

For example,

– with N = X(8) hence tcN = X(75), we obtain psK+(X8, X75, X8) corresponding to pK(X6, X2) = K002 as above.

– with N = X(69) hence tcN = X(76), we obtain psK+(X69, X76, X69) corresponding to pK(X6, X75) = K968.