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K735

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X(3), X(182), X(3098), X(8666), X(8715)

vertices of the CircumTangential triangle

vertices of the CircumNormal triangle

infinite points of K002

K735 is a central cubic with center O, meeting the circumcircle at the vertices of CircumTangential and CircumNormal triangles.

It has three real asymptotes which are the parallels at O to those of the Thomson cubic K002.

It meets K002 again at six (real or not) points which lie on the rectangular hyperbola (H) passing through X(3), X(54), X(140), X(1511), X(2574), X(2575).

K735 is the CircumTangential isogonal transform of K078, a stelloid with asymptotes parallel to those of K003.

The CircumNormal isogonal transform of K735 is also a stelloid namely K736.

Generalization

For any point P = u : v : w not lying at infinity, one can find a central cubic with center O, meeting the circumcircle at the vertices of CircumTangential and CircumNormal triangles and passing through the infinite points of pK(X6, P).

Its equation is given

All these cubics form a net.

Now, if P and Q are two points not lying on the circumcircle and not collinear with O then the central cubic K(P, Q) with center O, meeting the circumcircle at the vertices of CircumTangential and CircumNormal triangles, passing through P, Q and their reflections P', Q' about O meets the line at infinity at the same points as pK(X6, S) where S is the intersection of the lines L(P), L(Q) passing through the CircumTangential isogonal conjugates of P and P', Q and Q' respectively. The general equation is huge and will not be given here.

K(P, Q) and pK(X6, S) meet again at six (real or not) finite points lying on a same rectangular hyperbola H(P, Q).

Under CircumTangential and CircumNormal isogonal conjugations, K(P, Q) is transformed into two stelloids both with asymptotes parallel to those of the McCay cubic K003. Each stelloid is the reflection about O of the other.

With P = X(182) and Q = X(8666), we obtain K735 and then P' = X(3098), Q' = X(8715), S = X(2), H(P, Q) = (H) as above. The two stelloids already mentioned above are K078 and K736.

K735a

In the opposite figure, P = X(6), Q = X(2),

hence P' = X(1350), Q' = X(376).

S is the intersection of the lines X(30)X(141) and X(511)X(8667) with SEARCH = -32.3886613230438.

Note that the six points mentioned above are all real in this figure.