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X(3), X(40), X(376), X(7709)

vertices of the CircumNormal triangle

infinite points of K003

A'B'C' reflection of the Thomson triangle about O

other points below

K736 is the CircumNormal isogonal transform of K735 which is itself the CircumTangential isogonal transform of K078.

K736 is a stelloid with three real asymptotes parallel to those of the McCay cubic K003 at X = X(10304). X is the reflection of X(3524) about X(3) and lies on the Euler line.

The tangents at A', B', C' concur at X(376) hence K736 is a pK in A'B'C' and it must meet the sidelines of A'B'C' again at three points which are the vertices of a triangle perspective with A'B'C' at a point on the cubic which turns out to be O. It follows that K736 is the pK with pivot O, isopivot X(376) in A'B'C' but since X(376) is the orthocenter of A'B'C', K736 is the McCay cubic for A'B'C'. X is the centroid of A'B'C' since X(3524) is that of the Thomson triangle. Finally, since the McCay cubic in the Thomson triangle is K078, we obtain that K736 is the reflection of K078 about O. Hence, it contains the reflection of X(5373) about O and more generally, the reflections of all the points mentioned in the page K078.

K736 also contains the vertices of the pedal triangle of X(376) with respect to the CircumNormal triangle.

K736 is a member of the pencil of cubics generated by K725 and K735. This pencil also contains a circumcubic passing through X(3), X(4).

***

K003 and K736 generate a pencil of stelloids with radial center on the Euler line.

This pencil contains two decomposed cubics, namely :

• the union of the altitudes of the CircumNormal triangle, with radial center O,

• the union of the line at infinity and the rectangular hyperbola with center X(1511), homothetic to the Jerabek hyperbola, passing through X(3), X(54), X(110), X(182), X(1147), X(1385), X(2574), X(2575), etc. The radial center is X(30).

Every undecomposed stelloid passes through the vertices N1, N2, N3 of the CircumNormal triangle, the infinite points of K003, and O counted three times since its polar line is the Euler line and its polar conic is the Stammler hyperbola.

Let S(P) be the stelloid that meets the Euler line again at P, which is obviously the tangential of O. Hence S(H) = K003 and S(X376) is K736.

S(P) meets the circumcircle (O) again at three points R1, R2, R3 that lie on a nK0(X6, R) where R is a point on the line GK. The antipodes Q1, Q2, Q3 on (O) of these points lie on pK(X6,Q), where Q is the reflection of P in O.

When P traverses the Euler line, the line PR envelopes an ellipse passing through X(4), X(32), X(1506), X(6781), X(7736). The tangents at X(4), X(7736) are the Euler line and the line GK respectively. The center is the midpoint of X(4), X(6781), also on the lines {114,754}, {132,186}, SEARCH = -5.90645593709317.

Other cubic in the pencil : K1270.