HYACINTHOS 28969

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

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

N1, N2, N3 = the NPC centers of PNbNc, PNcNa, PNaNb, resp.

Which is the locus of P such that the reflections of:

1. PNa. PNb, PNc

2. PN1, PN2, PN3

3. NaN1, NbN2, NcN3

 

in BC, CA, AB, resp. are concurrent?

 

[César Lozada]:

 

1)      Locus={Linf} ∪ {circumcircle} ∪ { K005 (Napoleon-Feuerbach cubic)}

ETC pairs (P,Q1(P)) for P ∈ K005: (1,36), (3,3), (4,110), (5,6150), (17,30559), (18,30560), (54,30)

 

Q1( X(61) ) = X(3)X(13) ∩ X(62)X(1337)

= (SB+SC)*((15*R^2-2*SA+3*SW)*S^2+sqrt(3)*(S^2+SA^2+4*SB*SC-(21*SA-3*SW)*R^2)*S-3*(R^2-SA+2*SW)*SA*SW) : : (barys

= lies on these lines: {3, 13}, {62, 1337}

= midpoint of X(8172) and X(16965)

= [ -4.4928056704912800, -3.8879342511705630, 8.4059138883292070 ]

 

Q1( X(62) ) = X(3)X(14) ∩ X(61)X(1338)

= (SB+SC)*((15*R^2-2*SA+3*SW)*S^2-sqrt(3)*(S^2+SA^2+4*SB*SC-(21*SA-3*SW)*R^2)*S-3*(R^2-SA+2*SW)*SA*SW) : : (barys)

= lies on these lines: {3, 14}, {61, 1338}

= midpoint of X(8173) and X(16964)

= [ 6.1080011092578240, 6.1201598565486650, -3.4154466999760020 ]

 

2)      Locus = {Linf} ∪ {nasty degree 22 through ETC’s 5, 13, 14, 110}

ETC pairs (P,Q2(P)): (5, 6150), (13,15), (14,16)

 

Q2( X(110) ) = X(110)X(924) ∩ X(476)X(12092)

= (SB+SC)*(SA-SB)*(SA-SC)*(S^2-27*R^4-2*R^2*(3*SA-8*SW)+SA^2-2*SB*SC-2*SW^2) : : (barys)

= lies on these lines: {110, 924}, {476, 12092}, {933, 1291}, {2071, 10628}

= [ -17.4302611072831800, 6.3402837081745140, 7.2959731949942600 ]

 

3)      Locus = {Linf} ∪ {horrendous degree 25 through ETC’s X(5) }

Q3(X(5)) = X(6150)

 

César Lozada