POINTS ON THE McCAY CUBIC (K003) - 2

[APH = Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P

Denote

Nbc, Ncb = the NPC centers of PBC', PCB', resp.

Nca, Nac = the NPC centers of PCA', PAC', resp.

Nab, Nba = the NPC centers of PAB', PAC', resp.

 

Which is the locus of P such that  Nbc, Ncb, Nca, Nac, Nab, Nba lie on a conic?

Note: I have read that for P = H the six points are conconic, but I do not remember where.

***lapsus memoriae***
For which P's  the Nbc, Ncb, Nca, Nac, Nab, Nba are concyclic?

 

[César Lozada]:

Locus for points on a conic: A quintic, circumscribing triangles {ABC, ABC-X3 reflections, infinite-altitude, X3-ABC reflections} . 

The only known ETC center on it is X(3).

For P=X(3) the conic is an ellipse with center X(140) and perspector:
Z = ISOGONAL CONJUGATE OF X(22233)
= (8*a^8-(29*b^2+35*c^2)*a^6+(39*b^4+29*b^2*c^2+54*c^4)*a^4-(b^2-c^2)*(23*b^4-6*b^2*c^2-35*c^4)*a^2+(5*b^2-8*c^2)*(b^2-c^2)^3)*(8*a^8-(35*b^2+29*c^2)*a^6+(54*b^4+29*b^2*c^2+39*c^4)*a^4-(b^2-c^2)*(35*b^4+6*b^2*c^2-23*c^4)*a^2+(8*b^2-5*c^2)*(b^2-c^2)^3) : :

= lies on these lines: {11539, 40684}

= isogonal conjugate of X(22233)

= intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(140)}} and {{A, B, C, X(3), X(11539)}}

= [ 3.6509898050992220, 2.4513665130743210, 0.2584923705025162 ]

No other remarkable points were found on this ellipse.

> ***lapsus memoriae***
> For which P's  the Nbc, Ncb, Nca, Nac, Nab, Nba are concyclic?

Not such lapsus. If you ask for locus for 4 points to be concyclic, calculus leads to a single equation. But if you ask for locus such that 6 points are concyclic, calculus leads to three lateral equations whose intersections are the desired locus.  Most of the times, these equations can’t be algebraically solved for degree>=3.

 

Locus for circularity:  P  ∈ {intersections of three lateral cubics, Ka, Kb, Kc}

 

Ka = -(2*a^6+a^2*b^2*c^2+7*a^2*b^4+3*b^4*c^2-c^6+4*a^2*c^4-5*a^4*c^2-7*a^4*b^2-2*b^6)*c^2*x^2*y+b^2*(-6*b^2*c^2-7*a^2*b^2-5*a^2*c^2+2*a^4+3*c^4+3*b^4)*c^2*x^2*z-(9*a^2*b^4-2*b^6+c^6+a^2*c^4-5*a^4*c^2-6*a^2*b^2*c^2+5*b^4*c^2-4*b^2*c^4-10*a^4*b^2+3*a^6)*c^2*x*y^2-2*(5*a^2*b^4-b^2*c^4-2*a^2*b^2*c^2-3*a^4*b^2-b^6+2*b^4*c^2+a^2*c^4-2*a^4*c^2+a^6)*c^2*x*y*z+b^2*(-b^6+4*b^4*c^2+4*a^2*b^4-5*b^2*c^4-8*a^2*b^2*c^2-5*a^4*b^2+2*c^6-4*a^4*c^2+2*a^6)*x*z^2-a^2*(c^4-3*a^2*b^2+2*b^4+a^4-2*a^2*c^2-3*b^2*c^2)*c^2*y^3-a^2*(-11*b^2*c^2-9*a^2*b^2+7*b^4+4*a^4-8*a^2*c^2+4*c^4)*c^2*y^2*z+a^2*(a^6-2*a^4*b^2-4*a^4*c^2+a^2*b^2*c^2+a^2*b^4+5*a^2*c^4+7*b^2*c^4-2*c^6-5*b^4*c^2)*y*z^2+b^2*a^2*(2*c^4-3*a^2*c^2-3*b^2*c^2+a^4-2*a^2*b^2+b^4)*z^3 = 0

and cyclically Kb, Kc from Ka.

 

Their intersections are 7 real points (P1..P7) and 2 imaginary points. Three of them can be easily expressed:

P1 =2*a^2*b^2 : -b^2*(a^2+b^2-c^2) : c^4-3*a^2*b^2+2*b^4+a^4-2*a^2*c^2-3*b^2*c^2

and P2, P3  obtained cyclically from P1. But these degenerate the circle.

 

Conjectures (based on numerical analysis):

·There is just one center P* such that the nine-point-centers are concyclic on a circle (O*)

·P* lies on these cubics: K003 (McCay cubic again), K762, K849, K854

·O* lies on K258

 

ETC-(6-9-13)-search numbers for points:

1) {-8.67698625990118, 2.06523164148807, 6.21565161929319}

2) {-5.13431209747574, -2.47207619508091, 7.72178435425997} = P1

3) {0.0222518774850184, 0.0255746642599759, 3.61268884781131} = P*

4) {0.151423470638859, -16.0684952680259, 14.6951196040921} = P3

5) {3.92996728448760, 9.78031284777500, -4.94415239016182} = P2

6) {4.27283085639944, -10.4257496351357, 8.88641537250938}

7) {31.5113551028021, 31.9147470363133, -32.9978627445258}

 

8) {0.686609686609687 - 0.269678850511663 i, 0.566951566951567 + 0.471937988395411 i, -0.709401709401709 - 0.202259137883748 i}

 

9) {0.686609686609687 + 0.269678850511663 i, 0.566951566951567 - 0.471937988395411 i, -0.709401709401709 + 0.202259137883748 i}

 

César Lozada