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X(2), X(4), X(468), X(525)

other points below

CL026 is the class of cubics nK0(W, W) that contains a subclass of interesting cubics having a pencil of circular polar conics when W lies on the orthic axis. In other words one can find a line of points having circular polar conics which we call the circular line of the cubic. Any such cubic has three concurring asymptotes at G : one is real and two are imaginary conjugates.

There are generally two other cubics nK0 with the same pole W and having the same property. Their roots R1, R2 are two points of K600 collinear with G.

Construction of K600 :

Let W be a point on the orthic axis and let L be the line passing through G and the barycentric quotient W ÷ H on the line at infinity.

The circle with diameter HW meets L at the two required points R1, R2.

Properties of K600 :

K600 is a circular cubic with focus X(132). Its real infinite point is X(525).

Apart the points above, K600 contains :

• the intersections P1, P2 of the orthic axis and the Kiepert hyperbola. Their midpoint is X(1637) and the circle with diameter P1P2 contains the Fermat points X(13), X(14).

• Q1 = GP1 /\ HP2 and Q2 = GP2 /\ HP1. These points lie on the line through X(6), X(67), X(125), etc, and on the circle (C) passing through X(115), X(132), X(381), X(468), X(1637). The four points P1, P2, G, H form an orthocentric quadrilateral with diagonal triangle Q1 Q2 X(468). The polar conic of X(525) in the cubic is a rectangular passing through P1, P2, G, H and obviously X(525), X(1503). It follows that K600 is an isogonal pivotal cubic with respect to the triangle Q1 Q2 X(468) and the four points above are the centers of anallagmaty.

In particular, the inversion with pole G swapping H and X(468) leaves the cubic unchanged. The circle of inversion is the orthoptic circle of the Steiner inellipse.

Similarly, the inversion with pole H swapping G and X(468) also leaves the cubic unchanged. The circle of inversion is the polar circle hence K600 is invariant under orthoassociation.

• the antipode X of X(132) on the circle which is also the fourth vertex of the rectangle G X(132) X(1637) X. X is the point where the cubic intersects its real asymptote.

Example :

The most interesting example is obtained with W = X(1990). The three corresponding cubics are nK0(X1990, X1990), nK0(X1990, X468), nK0(X1990, X4) = K393 with respective circular lines the Euler line, the line HK, the orthic axis.

A special case :

When W is one the points P1, P2 above two of the three cubics coincide hence the only corresponding cubics are nK0(Pi, Pi) and nK0(Pi, Qi).


See K604, Q087, CL044 for other cubics with similar properties and also Table 47.