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X(1), X(3), X(4), X(1075), X(1745), X(3362), E(412)=X(1075)*

Ia, Ib, Ic vertices of excentral triangle

vertices of the circumnormal triangle

more points and details at the bottom of the page

The McCay cubic is the isogonal pK with pivot O = X(3). See Table 27. It is sometimes called Griffiths cubic. It is a member of the classes CL006, CL009, CL021, CL024 of cubics. It is the only isogonal pK60+ and the three asymptotes concur at G. These asymptotes meet the cubic at three finite points which lie on the line homothetic of the Brocard line OK under h(G,2/3).

Since K003 is an equilateral cubic, the polar conic of any point of the plane is a rectangular hyperbola. The locus of centers of the polar conics of the points on the McCay cubic is Q048, the McCay sextic.

Locus properties :

  1. Locus of point P such that the polar lines (in the circumcircle) of P and its isogonal conjugate P* are parallel. Equivalently, locus of P such that P, P*, iP (inverse of P in the circumcircle) are collinear. Compare this with K105, property 1.
  2. Locus of point P such that the polar lines (in the inscribed ellipse with center O, perspector X(69)) of P and its isogonal conjugate P* are parallel.
  3. Denote by PaPbPc the pedal triangle of point P and by T1, T2, T3, T4, T5, T6 the areas of triangles PBPa, PPaC, PCPb, PPbA, PAPc, PPcB respectively. The locus of point P such that T1*T3+T3*T5+T5*T1 = T2*T4+T4*T6+T6*T2 is the McCay cubic.
  4. Locus of point P such that the sum of line angles (BC,AP)+(CA,BP)+(AB,CP)=pi/2 (mod. pi). Compare this with K024, property 6.
  5. Locus of point P whose pedal circle is tangent to the nine point circle. See a generalization in table 22.
  6. Locus of point P whose pedal and circumcevian triangles are perspective and even homothetic. The perspector is a point of the Lemoine cubic. See also CL024.
  7. Locus of point P such that the antipedal triangle of P and the circumcevian triangle of its isogonal conjugate are perspective (together with the line at infinity and the circumcircle). These triangles are in fact homothetic.
  8. Denote by PaPbPc the circumcevian triangle of point P and by QaQbQc the triangle bounded by the Simson lines of Pa, Pb, Pc. PaPbPc and QaQbQc are homothetic if and only if P lies on the McCay cubic.
  9. PaPbPc and QaQbQc have the same area if and only if P lies on the McCay cubic.
  10. It is known that the shapes of repeated pedal triangles with respect to a fixed point P recur with period 3. The McCay cubic is the locus of P for which the third pedal is homothetic to the reference triangle.
  11. let P be a point. The parallels at A to PB, PC meet BC at Ab, Ac respectively and Oa is the circumcenter of AAbAc. Define Ob, Oc similarly. The triangles ABC and OaObOc are perspective if and only if P lies on the McCay cubic (together with the line at infinity and the circumcircle) (from Hyacinthos #5618)
  12. Locus of point P such that the pedal triangles of P and its isogonal conjugate P* are orthologic (together with the line at infinity and the circumcircle). See also CL021.
  13. Locus of point P such that the circumcevian triangle of P and ABC are orthologic. The locus of the orthology centers is Q046, the McCay butterfly.
  14. Locus of point P such that the reflections of P in the sidelines of the circumcevian triangle of P form a triangle perspective to ABC (together with the circumcircle).
  15. Locus of point P such that there is an isogonal pK with asymptotes parallel to the cevian lines of P. See CL009.
  16. (generalization of 12) Let t be a real number and P, Q two isogonal conjugates. PaPbPc, QaQbQc are the homothetic under h(P,t), h(Q,t) of the pedal triangles of P, Q respectively. PaPbPc and QaQbQc are orthologic if and only if P and Q lie on the McCay cubic (together with the line at infinity and the circumcircle). More generally, when P and Q are two isoconjugates under an isoconjugation with pole W, the locus is a member of the class CL021 of cubics (together with the line at infinity, its W-isoconjugate and the trilinear polar of the isotomic conjugate of the isogonal conjugate of W).
  17. Locus of point P such that the pedal and antipedal triangles of P are orthologic (together with the line at infinity and the circumcircle).
  18. Locus of point P such that the circumcenters of the pedal and antipedal triangles of P are collinear with P or O (together with the line at infinity). If we replace circumcenters by centroids, orthocenters, symmedian (Lemoine) points, nine-point centers, we obtain Q017, Q018, Q019, Q020 respectively.
  19. The perpendicular at P to BC meets the line AP* at A'. La is the parallel at A' to AP. Define B', C', Lb, Lc similarly. These lines La, Lb, Lc concur if and only if P lies on the union of the McCay cubic and the orthocubic. Now, if La is the parallel at A' to BC and if Lb, Lc are defined likewise, the lines concur if and only if P lies on the quartic Q023 (Floor van Lamoen and friends, Hyacinthos #5360 & sq.).
  20. Locus of point P such that the reflections of P in the sidelines of ABC form a triangle perspective to the circumcevian triangle of P (together with the circumcircle). The locus of the perspector is Q047, the inversive image of the McCay cubic in the circumcircle.
  21. Locus of point P such that the circumcevian and pedal triangles of P are orthologic (together with the line at infinity and the circumcircle).
  22. Locus of point P such that the circumcevian triangles of P and ABC are orthologic (together with the line at infinity and the circumcircle). The locus of the orthology centers is the McCay cubic itself and Q046, the McCay butterfly.
  23. Locus of point P such that the antipedal and reflection triangles of P are orthologic (together with the line at infinity and the circumcircle).
  24. Locus of point P such that the circumcevian and reflection triangles of P are orthologic (together with the line at infinity and the circumcircle).
  25. Locus of point P such that the circumcevian triangle of P and the triangle formed by the reflections of P in A, B, C are orthologic (together with the line at infinity and the circumcircle).
  26. Locus of point P such that the circumanticevian and reflection triangles of P are parallelogic.

Points on K003

A, B, C where the tangents are the altitudes

I, Ia, Ib, Ic where the tangents pass through O

O (pivot) where the tangent is the Euler line

H (isopivot) where the tangent pass through X(51) and X(1075)

X(1075) = O/H (cevian quotient)

E(412) = X(1075)* = isogonal conjugate of X(1075)

X(1745) the third point on IH and its extraversions on the lines HIa, HIb, HIc

the isogonal conjugates of these four points

Oa, Ob, Oc (cevians of O) where the tangents pass through X(1075)

N1, N2, N3 vertices of the circumnormal triangle

N1*, N2*, N3* their isogonal conjugates at infinity

H1, H2, H3 projections of H on the sidelines of N1N2N3. These points lie on the bicevian conic C(G,O) with center X(140) passing through X(125), X(1511), X(2972) and obviously Oa, Ob, Oc.

H1*, H2*, H3* their isogonal conjugates

foci of the inconic with center O, perspector X(69), passing through X(125), X(1565), X(2968). These four points lie on the parallels at O to the asymptotes of the Jerabek hyperbola. See property 2 above.

Inscribed equilateral triangles

K003 contains nine points on the cevian lines of the vertices of the circumnormal triangle and their nine isogonal conjugates. The former nine points lie on the Apollonius circles and form three equilateral triangles with sidelines perpendicular to those of the circumtangential triangle. These points are three by three collinear with the isodynamic points. Compare this configuration and the analogous configuration with the Kjp cubic.
K003Apollonius

More generally, there are infinitely many equilateral triangles inscribed in the McCay cubic. Their centers lie on the McCay equilateral quintic Q065.
K003CICO

In particular, the two circles passing through H, X(74) and one of the intersections X(1113), X(1114) of the Euler line and the circumcircle meet the McCay cubic at the vertices of two such triangles.

These circles are orthogonal and centered at O1, O2 on the perpendicular bisector of HX(74) and on the axes of the inconic with center O (these axes are the parallels at O to the asymptotes of the Jerabek hyperbola). O1, O2 also lie on the circle passing through O, H, X(74). Each circle contains two foci of this inconic (these four foci obviously on the McCay cubic).

The two triangles have their sidelines perpendicular and their vertices collinear with H. Since H is the isopivot of the McCay cubic, the vertices of one triangle are the O-Ceva conjugates of the vertices of the other triangle.

Naturally, the isogonal conjugates of these six vertices also lie on the McCay cubic.

K003 and the Steiner ellipse

K003 meets the Steiner ellipse at A, B, C and three other points M1*, M2*, M3* which are the isogonal conjugates of the common points M1, M2, M3 of the Lemoine axis and K003.
K003Steiner

These points M1, M2, M3 are the perspectors of ABC and A'B'C', the only triply bilogic triangle inscribed in the circumcircle (Jean-Pierre Ehrmann, Hyacinthos #14350).

This means that ABC and A'B'C' are triply perspective and orthologic. M1*, M2*, M3* are three of the six centers of orthology. The three other lie on Q046, the McCay butterfly.