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X(3), X(4), X(15), X(16), X(17), X(18), X(54), X(140), X(195), X(1263)

K469 is a remarkable circumcubic that contains the isodynamic points X(15), X(16) and the Napoleon points X(17), X(18). See the related circular cubic K468 and K471.

The isogonal transform of K469 is K470.

K469 and K470 generate a pencil of cubics containing the Orthocubic K006 and the non-pivotal isogonal cubic nK(X6, X110 x X140, X3).


K469 is the pivotal cubic with pivot H and isopivot X(54) with respect to the triangle T whose vertices are X(15), X(16), X(140).

The cevian triangle of the pivot is X(3)X(17)X(18).

Any two points collinear with H are therefore isoconjugate points with respect to T. This is the case of X(3) and X(140), X(15) and X(17), X(16) and X(18), X(195) and X(1263).

The tangents at H, X(15), X(16), X(140) pass through X(54).

More generally, let us denote by :

• (K) and (H) the cubic K469 and its hessian,

• P a point on (K) which is not a flex, (P = H on the figure),

• Q its tangential, the intersection of the polar lines of P in (K) and (H), (Q = X(54) on the figure),

• (PK), (QK) the polar conics of P and Q in (K),

• (PH) , (QH) the polar conics of P and Q in (H),


(QK) and (QH) intersect at P and three other (blue) points P1, P2, P3, (these are X(15), X(16), X(140) on the figure),

(PK) and (PH) intersect at four (magenta) points I0, I1, I2, I3 whose diagonal triangle is P1P2P3,

(K) is a pivotal cubic with pivot P, isopivot Q with respect to P1P2P3 and I0, I1, I2, I3 are the fixed points of the isoconjugation.

Note that the lines QP1, QP2, QP3 (which are the tangents to (K) at P1, P2, P3) are the polar lines of P in the three prehessians of (K).