HYACINTHOS 29221

[Antreas P. Hatzipolakis]:

 

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

Denote:

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

The perpendicular from Na to BC intersects AP at A*

The perpendicular from Nb to CA intersects BP at B*

The perpendicular from Nc to AB intersects CP at C*  

For which P's:

1. ABC, A*B*C* are homothetic?
N is such a point. Others?

2. A'B'C', A*B*C* are homothetic?
I is such a point. Others?



[César Lozada]:

1)      Locus = {Linf} ∪ {Euler line}

The homothetic center is P

 

2)      Locus={Linf} ∪ {McCay cubic K003}

ETC pairs (P,H2(P) ): (1,1319), (3,3), (4,4)

 

H(X(1075)) = X(3)X(1075) ∩ X(14249)X(18400)

= (S^4-2*(16*R^4+SB*SC-SW^2)*S^2-(4*R^2-SW)*(16*(36*R^2+8*SA-23*SW)*R^4+4*(2*SA^2-18*SA*SW+19*SW^2)*R^2+(6*SA^2+2*SA*SW-5*SW^2)*SW))*SB*SC : : (barys)

= lies on these lines: {3, 1075}, {14249, 18400}

= [ -1.8268482753765900, -1.2369702975809820, 5.3401892765604080 ]

 

César Lozada