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X(1), X(3), X(4), X(1075), X(1745), X(3362), X(13855) Ia, Ib, Ic vertices of excentral triangle vertices of the circumnormal triangle foci of the inconic with center O, perspector X(69) imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola more points and details below 

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. See also Table 63. 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). The McCay cubic is an example of stelloid. See also K024, Table 22 and Table 51. 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. In other words, the Psi transform of K003 is Q048. More on polar conics below. The symbolic substitution SS{a > √a} transforms K003 into K977. Locus properties (see also Table 7)


Locus properties related to Simson lines :


Related papers : 

Miscellaneous properties 

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) X(13855) = 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 

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola 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. 

Seven special points on K003 (RescassolSkoubidou points) 

These points solve two questions asked independently and almost simultaneously during January 2015 in lesmathématiques.net (in French, thread Droites parallèles 20151) and ADGEOM (thread Find a point?). In both cases, the solution was given by JeanPierre Ehrmann. Here is a short (slightly rephrazed) summary. Question 1 : find P such that the pedal circles of P with respect to ABC and the circumcevian triangle of P are identical. Question 2 : the perpendiculars at P to AP, BP, CP meet the sidelines AB and AC, BC and BA, CA and CB respectively at six points on a same conic. Find P such that this conic is a circle. JeanPierre Ehrmann proved that there are 7 (not always real) such points, interior to the circumcircle and lying on K003. These are not ruler and compass nor conic constructible. See the mentioned threads for further details. 

Obviously, the isogonal conjugates of these points are seven other points on K003. In the figure above, there is one and only one real point P. For an almost "flat" triangle ABC, one can find up to five real such points P. Recall (see property 6) that the pedal triangle PaPbPc of P and the circumcevian triangle A'B'C' of P are homothetic. 

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. A construction is given below. Compare this configuration and the analogous configuration with the Kjp cubic. 

More generally, there are infinitely many equilateral triangles inscribed in the McCay cubic since it is a stelloid with radial center G. Their centers lie on the McCay equilateral quintic Q065. 

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 OCeva 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. These points M1, M2, M3 are the perspectors of ABC and A'B'C', the only triply bilogic triangle inscribed in the circumcircle (JeanPierre 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. 

K003 : polar conics, hessian and "negative"hessian Recall that the polar conic C(M) of any point M with respect to K003 is a rectangular hyperbola. The hessian of K003 is K048. It is the locus of point M whose polar conic in K003 degenerates into two perpendicular lines secant at N which also lies on K048. The mapping F : M > N is the product of the reflexion about an axis of the Steiner inellipse and the inversion with respect to the circle centered at G passing through the real foci of this same ellipse. C(M) is a bicevian conic if and only if M lies on a cubic (K) we call the negativehessian of K003. Indeed, K048 is given by the determinant of the hessian matrix of K003 (see Special Isocubics, §2) and then, when the signs in the diagonal are changed, we obtain another matrix whose determinant gives (K). Properties of (K) 

(K) meets the McCay cubic K003 at 9 points namely : • the in/excenters of ABC, each counted twice since the tangents are the same in both cubics and pass through O. The perspectors of these four bicevian conics are complicated. • X(1075), the X(3)Ceva conjugate of X(4), whose polar conic passes through the vertices A', B', C' of the cevian triangle of X(3). The other perspector is unlisted in ETC with 1st barycentric 1 / [a^2 (b^2 – c^2) SA^3] and SEARCH = –0.266974863185788. (K) also contains X(20), X(185). The polar conic of X(20) in K003 is the bicevian conic C(X2, X110) which is at the same time : • the complement of the Jerabek hyperbola, • the X(2)Ceva conjugate of the Brocard axis, • the polar conic of X(5) in K002. 

The polar conic of X(185) in K003 passes through X(4), X(281), X(1068) with two complicated perspectors. Note that the polar conics of X(3) in K048 and (K) are the same diagonal conic (C) passing through the in/excenters, X(661), X(896). (K) belongs to the pencil generated by K048 and the union of the cevian lines of X(3) hence (K) meets K048 at three triads of (not always all real) points. (K) meets the line at infinity at the same points as pK(X6, Z) where Z is the midpoint of X(20), X(185). The six remaining finite common points lie on the diagonal conic passing through the in/excenters and X(371), X(372), X(1707), X(1724), X(1754). 

K003 : prehessians K003 has three real prehessians P1, P2, P3. In other words, the hessians of P1, P2, P3 are K003. Naturally, K003, its hessian and these three curves share the same nine inflexion points. They belong to a same pencil of cubics called syzygetic pencil of cubics. 

These cubics Pi are associated to three new mappings Fi defined as follows. Fi maps any point M to the center of the polar conic of M in Pi. Fi is an involution of the plane that leaves K003 globally unchanged. Note that the polar conics of all the points of the Brocard axis in Pi are rectangular hyperbolas. This shows that the prehessians Pi share the same orthic line, namely the Brocard axis. The polar conics of the centroid G in P1, P2, P3 are the Apollonian circles passing through A, B, C respectively. The three singular points of F1 are the vertices of the equilateral triangle AaAbAc inscribed in the AApollonian circle. See above. This latter circle is actually the image of the Brocard axis under F1. Recall that these points Aa, Ab, Ac, etc, being singular must lie on the McCay cubic. 

More generally, F1 transforms any line through O into a conic passing through A and Aa, Ab, Ac and, in particular, the Euler line into a rectangular hyperbola passing through the Acevian Oa of O, the center Ωa of the AApollonius circle. The tangent at A is the altitude AH and the axes are the parallels at the midpoint of AΩa to the bisectors at A of ABC. This gives a construction of Aa, Ab, Ac. *** The poloconic of a line L in Pi is a conic tritangent to K003 at the images of the common points of L and K003 under Fi. If L is tangent to K003, the conic is a surosculating conic and is tangent to the McCay cubic at another point. For example : – if L is the tangent at O to K003 (the Euler line), the conics are surosculating at A, B, C and pass through Oa, Ob, Oc, – if L is the tangent at I to K003 (the line IO), the conics are surosculating at the excenters Ia, Ib, Ic and pass through A, B, C.
If L is an inflexional tangent at P to K003, the conic is a sextactic conic. The poloconic of L has a sextuple contact with K003 at Fi(P). *** 

Feuerbach theorem in K003 There are four lines (not always real) such that their poloconics with respect to one of the three prehessians Ωi are circles. In such case, these circles are tritangent to K003 and tangent to the corresponding Apollonius circle. This is the generalized Feuerbach theorem for the McCay cubic : the Apollonius circle is analogous to the nine point circle and the four circles are analogous to the in/excircles. The figure shows a configuration where the four circles are real and tangent to the BApollonius circle. 

Parallels to the asymptotes of K003 

If an equilateral triangle with center G is drawn with sidelines parallel to the asymptotes of K003 then these sidelines meet K003 on the line at infinity and at six other points lying on a same circle whose center Ω lies on the perpendicular at G to the Brocard axis. When the sidelines concur at G, this circle splits into the line at infinity and its satellite which is parallel to the Brocard axis. This line is the homothetic of the Brocard axis under h(G, 2/3). 

If three parallels to the asymptotes of K003 are drawn through a point P then these parallels meet K003 on the line at infinity and at six other points lying on a same rectangular hyperbola (H1) whose center O1 is the midpoint of PO2 where O2 is the center of the rectangular hyperbola (H2) which is the polar conic of P in K003. These two rectangular hyperbolas have the same asymptotic directions therefore they are homothetic. Their two other (finite) common points lie on the polar line (L1) of P in (H1). This line (L1) is also the homothetic of the polar line (L2) of P in (H2) and in K003 under h(P, 2/3). When the parallels concur at G, this hyperbola splits into the line at infinity and its satellite. 

Two groups of points on K003 

As in property 29 above, consider six circles with same radius defined as follows. (C1) passes through B, C with center in the halfplane not containing A and (C'1) is its reflection about BC. The circles (C2), (C'2) and (C3), (C'3) are defined likewise. For i, j, k in {1,2}, define the point ijk as the radical center of three circles as Lemoine did. For example 111 is the radical center of (C1), (C2), (C3) and 112 is the radical center of (C1), (C2), (C'3). A group of eight (green) points is thus defined and these points all lie on the McCay cubic. If the common radius of the circles is t R (t > 0, R circumradius), then the trilinear coordinates of ijk are : cos A – i √(t^2 – sin^2 A) : cos B – j √(t^2 – sin^2 B) : cos C – k √(t^2 – sin^2 C). 

Collinearities related with sums modulo 3 Two points can be added (digit by digit) modulo 3 to obtain a certain sum. For example 111 + 112 = 220, 111 + 212 = 020, etc. The results thus obtained can be used to identify several collinearities on K003. • two points are collinear with O (hence isogonal conjugates) when the sum is 000. Examples : 111 and 222, 112 and 221, etc. • two points are collinear with A when the sum is 0jk. Examples : 111 and 211 (sum 022), 112 and 212 (sum 021), etc. • similarly, when the sum is i0k or ij0, we obtain points collinear with B or C. From this, we see that these eight points lie two by two on four groups of four lines passing through O, A, B, C respectively. It is clear that each sum of two distinct points must always contains at least one zero hence it remains to see the case when a sum contains two zeros. This gives the second group of six points labelled Yn, Zn on K003 with n in {1, 2, 3}. For example, Y1 is the intersection of the lines 111122, 212221 (sums 200, 100) and Z1 is the intersection of the lines 222211, 112121 (sums 100, 200). The triangles Y1Y2Y3, Z1Z2Z3 are perspective at H, and perspective to ABC at Y, Z on K003. Moreover, Y, Z, X(1075) are collinear on K003. Note that Y2Z3, Y3Z2 meet at A', the Acevian of O, and similarly with B', C'. 
