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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

table41a

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).

(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), cE(371) / equilateral pK

K142

see the page / equilateral

 

- / psK(X512, X2, ?)

K330

see the page / circular

K487

X(2), X(6), X(67), X(187), cE(371)

K331

see the page / circular

 

X(6) / circumcubic

K488

see the page

 

X(6), X(50), X(1989)

K732

see the page

 

- / nK(X6 x X1989, X265, ?)

K753

see the page / circular

K277

X(6), X(37), X(216), X(395), X(396), X(1989)