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X(2), X(15), X(16), X(511), X(1113), X(1114), X(7709), X(13414), X(13415)

foci of the orthic inconic or K-ellipse (inellipse with center K when the triangle ABC is acutangle)

S1, S2 defined in page K019. See also Table 62. These points are X(13414), X(13415)

Y orthogonal projection of G on the Lemoine axis, also on the lines {2,51}, {25,2076}

The McCay cubic is the only isogonal pK whose hessian cubic K048 is circular. The common points of the two cubics are obviously points of inflexion.

K048 is a focal cubic with singular focus G. The polar conic of the singular focus is the Parry circle. The real asymptote is parallel at X = X(7709) to the Brocard axis which is the orthic line of the cubic. X is the tangential of G.

K048 is the locus of contacts of tangents drawn from G to the circles passing through the isodynamic points X(15) and X(16).

A line passing through G intersects the Brocard axis at G'. The (Schoute) circle with centre G', orthogonal to the Parry circle, intersects this line at two points of K048.

See the related cubic K193 and also K019, K417, K418, K1257.

K048 is invariant under isogonal conjugation with respect to the triangle T with vertices X(2), X(15), X(16) and also under the involution Psi described in the page K018 and in the paper "Orthocorrespondence and Orthopivotal Cubics", §5. Indeed, these two involutions coincide for any point on K048. K048 is a Psi-cubic as in Table 60.

More precisely, K048 is a nK in T and meets the sidelines of T on the Brocard axis.

K048 is also invariant under the JS involution described in Table 62.



Let F be a point not lying on the sidelines of ABC, the line at infinity, the Brocard and Lemoine axes.

Consider two pencils of circles (F1) – resp. (F2) – with base points – resp. limit points – the isodynamic points X(15), X(16).

(F1) contains the Apollonius circles and (F2) contains the circumcircle (O) and the Brocard circle.

For any circle of (Fi), the locus of its common points with the polar of F is a focal cubic (Ki).

These two cubics (K1), (K2) have the same focus F and pass through X(15), X(16). They must meet at two other points which are the imaginary common points of (O) and the Lemoine axis.

The two remaining common points of (O) and (K1), (K2) lie on the line OF, the polar line of F in (O) respectively.

Note that (K1) and (K2) meet at F, X(15), X(16) with perpendicular tangents.

Properties of (K1)

The orthic line is the Brocard axis hence (K1) is always unipartite and contains X(511). The polar conic (C1) of F is the circle passing through F, X(15), X(16). The real asymptote is the homothetic of the Brocard axis under h(F,2) meeting the cubic at T1, the reflection of F in the center O1 of (C1).


Properties of (K2)

The orthic line is the Lemoine axis hence (K2) is always bipartite and contains X(512). The polar conic (C2) of F is the circle of (F2) passing through F. The real asymptote is the homothetic of the Lemoine axis under h(F,2) meeting the cubic at T2, the reflection of F in the center O2 of (C2).


Miscellaneous properties

The asymptotes of the two cubics meet at T3, the reflection of F in X(187), the common point of the Brocard and Lemoine axes.

The polar conics of X(511), X(512) in (K1), (K2) are two rectangular hyperbolas (H1), (H2) passing through X(187) and tangent at this point to the line F X(187). Each is the reflection in X(187) of the other.

(C1), (C2) meet at F and another point T4 on the line passing through T1 and T2.


Prehessians of (K1) and (K2)


Each cubic (K1), (K2) has three prehessians and one is always real, being a stelloid with radial center F.

More precisely,

• (K1) has only one real prehessian, the stelloid (P1),

• (K2) has three real prehessians, one being the stelloid (P2).

These stelloids (P1), (P2) which have the same radial center F – the common singular focus of (K1), (K2) – must intersect at three real points A', B', C' and six imaginary points, two by two on three lines which are the perpendicular bisectors of A'B'C', hence concurring at the circumcenter O' of A'B'C'. O' lies on the Brocard axis.

The tangents at A', B', C' to the two stelloids are two by two perpendicular.

A', B', C' form a group of pivots on both stelloids and F is the centroid of triangle A'B'C'. It follows that each stelloid meets the circumcircle of A'B'C' at the vertices of an equilateral triangle.

(P1), (P2) can be seen as the McCay and Kjp cubics for A'B'C'.

Indeed, when F = G, A'B'C' is the reference triangle ABC and these stelloids are precisely K003 and K024.