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X(2), X(6), X(187), X(216), X(395), X(396)

Z1 = X(2)X(187) /\ X(30)X(39) /\ X(32)X(381), etc

midpoints of ABC

U, V, W : their reflections in the corresponding feet of the orthic axis

Ωa, Ωb, Ωc see Table 41 and CL052

For any pole Ω different of K, there is a unique circular pivotal cubic pKc with pole Ω and for any Ω different of Ωa, Ωb, Ωc (see table 41 and CL052) there is a unique equilateral cubic pKe with pole Ω.

These two cubics pKc and pKe generate a pencil F of pivotal cubics with pole Ω and pivot P on the line passing through the pivots Pc and Pe of pKc and pKe. In particular, this line contains G/Ω (ceva conjugate) and pK(Ω, G/Ω) is a pK+.

This pencil contains a least one central cubic pK++ if and only if Ω lies on Q081 or on the Steiner inellipse.

More precisely :

– when Ω lies on the Steiner inellipse, one of the fixed points of the isoconjugation lies on the line at infinity and the pencil F contains a degenerate pK++ which is the union of the cevian lines of this infinite point.

– when Ω lies on Q081, the line PcPe contains G and F contains a generally proper pK++. The pK+ has its pivot on the complement of Q050.

The following table gives a selection of these cubics.

Ω

pKc

pKe

pK++

pK+

line of pivots

X(2)

K008

K092

medians of ABC

X(2)X(187)

X(187)

K043

 

K042

pK(X187, X187÷X316)

X(2)X(67)

X(216)

pK(X216, X3153)

K096

K044

pK(X216, X5)

Euler line

X(395)

K066b

K046b

pK(X395, X619)

X(2)X(14)

X(396)

K066a

K046a

pK(X396, X618)

X(2)X(13)