table41


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The points Ωa, Ωb, Ωc are the only points for which there are infinitely many equilateral pKs with pole one of these points. They lie on a large number of curves although they are not constructible with ruler and compass only. See also CL052 and Table 18.

Conics

Two conics

These three points lie on two conics (C1) and (C2) that meet at a fourth point Z, intersection of the lines X(5)X(1989) and X(6)X(30).

Z = X(14836) in ETC.

(C1) contains X(6), X(53), X(216) and X(1989).

(C2) contains X(395), X(396), X(523) and X(3003).

Cubics

These points Ωa, Ωb, Ωc lie on the PΩe–transforms of all the cubics that pass through the points Ua, Ub, Uc of Table 18.

The following table gives a selection of these cubics.

Cubics passing through Ua, Ub, Uc		Cubics passing through Ωa, Ωb, Ωc
cubic	centers / remark	cubic	centers / remark
K001	see the page / circular pK	K095	see the page / pK
K009	see the page / nodal	K486	X(6), X(53), X(2965) / nodal
K080	see the page / equilateral	K307	X(6), X(53), X(216), X(1249)
K092	see the page / equilateral	K485	X(2), X(11058), X(14537) / equilateral pK
K142	see the page / equilateral		- / psK(X512, X2, ?)
K330	see the page / circular	K487	X(2), X(6), X(67), X(187), X(11058)
K331	see the page / circular		X(6) / circumcubic
K488	see the page		X(6), X(50), X(1989)
K732	see the page		- / nK(X11060, X265, ?)
K753	see the page / circular	K277	X(6), X(37), X(216), X(395), X(396), X(1989)