HYACINTHOS 24873

[Antreas P. Hatzipolakis]:

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

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

N1, N2, N3 = the reflections of Na, Nb, Nc in BC, CA, AB, resp.

Which is the locus of P such that

1. A'B'C', NaNbNc

2. A'B'C', N1N2N3

are perspective?

 

[César Lozada]:

1)      Locus = {circumcircle} \/ { uncathalogued excentral circumcubic  pK(6,550) CyclicSum[b*c*(3*S^2-5*SB*SC)* u*(v^2-w^2]=0 (trilinears) through ETC’s 1,3,4,550}

 

(P,  perspector=Z1(P) ): (1, 5506), (3,3), (4,5)

 

Z1(X(550) = (386*cos(2*A)+26*cos(4*A)+63)* cos(B-C)+(72*cos(A)-16*cos(3* A))*cos(2*(B-C))+(2*cos(2*A)- 16)*cos(3*(B-C))+12*cos(5*A)+ 84*cos(A)-316*cos(3*A)) :: (trilinears)

= [ 0.229677805688296, -0.66457898109314, 3.994752481577337 ]

 

For P on the circumcircle, Z1(P) = reflection of P-in-O

 

2)       Locus = {circumcircle} \/ { Neuberg cubic }

ETC-pairs (P, Z2(P)=perspector): (1,3336), (3,3)

 

Z2(H) = (4*cos(2*A)+1)*cos(B-C)+2*cos( A)*cos(2*(B-C))+2*cos(A)+3* cos(3*A) : : (trilinears)

= 2*(R^2-SW)*X(54)+SW*X(69)

= On lines: {2,1493}, {3,2889}, {4,539}, {5,195}, {20,1154}, {24,9925}, {54,69}, {146,382}, {155,2914}, {193,6152}, {376,10619}, {524,7512}, {1209,5067}, {1992,9972}, {2892,8549}, {3528,7691}, {3574,3855}, {3832,6288}, {4309,6286}, {4317,7356}, {5070,8254}, {6193,6242}, {6639,9716}, {9705,10274}

= reflection of X(i) in X(j) for these (i,j): (2888,195), (3519,1493)

= anticomplement of X(3519)

= {X(1493), X(3519)}-Harmonic conjugate of X(2)

= [ -58.579386480671160, 47.47758595741437, -2.191793574454266 ]

 

The general expression for Z2(P) and its locus are rather complicated.

 

Regards,

César Lozada