HYACINTHOS 25053

[Antreas P. Hatzipolakis]:

 

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

Denote:

Ab, Ac = the reflections of B', C' in PA'

A2, A3 = the orthogonal projections of A on PAb, PAc, resp.

N1 = the NPC center of AA2A3

Similarly N2, N3.

Which is the locus of P such that ABC, N1N2N3 are orthologic?

The Euler line?

And which are the loci of the orthologic centers as P moves on the Euler line?

 

Generalization:

Denote:

Pa, Pb, Pc = same points on the Euler lines of AA2A3, BB3B1, CC1C2, resp.

ABC, PaPbPc are orthologic  ??.

 

[César Lozada]:

 

> Which is the locus of P such that ABC, N1N2N3 are orthologic? The Euler line?

Yes.

 

> And which are the loci of the orthologic centers as P moves on the Euler line?

Locus Zan=(ABC,N1N2N3) = a circum-conic. Center and perspector rather complicated. No ETC centers on it.

 

Locus Zna=(N1N2N3,ABC) = line {30, 1216, 5907, 10627 }.

If OP=t*OH then X(1216)*Zna = ((9*R^2-2*SW)*t-3*R^2+2*SW)/( 2*R^2) * X(1216)X(5907)

 

> Pa, Pb, Pc = same points on the Euler lines of AA2A3, BB3B1, CC1C2, resp. ABC, PaPbPc are orthologic  ??.

If P is on the Euler line of ABC and Pa,Pb,Pc are as described, then ABC and PaPbPc are always orthologic.

 

The locus of Zap(P,Pa) = (ABC, PaPbPc) is a conic not easy for finding further relations

 

The locus of Zpa(P,Pa) = (PaPbPc, ABC) is a variable line L(P)  through the fixed point:

 

Q = 2*a^10-3*(b^2+c^2)*a^8-(b^2-c^ 2)^2*a^6+(b^4-c^4)*(b^2-c^2)* a^4+(b^2-c^2)^2*(3*b^4+2*b^2* c^2+3*c^4)*a^2-2*(b^4-c^4)*(b^ 2-c^2)^3 : : (barycentrics)

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

= On lines: {4,51}, {5,1495}, {125,3575}, {184,7507}, {265,5446}, {373,7544}, {382,3581}, {546,6346}, {569,7564}, {1204,1853}, {1216,6288}, {1514,3861}, {3153,5907}, {3574,6146}, {3853,10113}, {5650,6643}, {6759,7547}, {7577,10282}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4,11550,11381), (6288,7574,1216)

= [ 3.985229021807456, 4.09359339501316, -1.032698186243557 ]

 

César Lozada