HYACINTHOS 25327

[Antreas P. Hatzipolakis]:

 

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

Denote:

A", B", C" = the orthogonal projections of A, B, C on PA', PB', PC', resp.

Ab, Ac = the orthogonal projections of A" on BO, CO, resp.
Bc, Ba = the orthogonal projections of B" on CO, AO, resp.
Ca, Cb = the orthogonal projections of C" on AO, BO, resp.

Na, Nb, Nc = the NPC centers of A"AbAc, B"BcBa, C"CaCb, resp.

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

The Euler line?

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

 

[César Lozada]:

 

> The Euler line?

Yes.

 

Za = ABC->NaNbNc: The locus is the circum-conic CyclicSum[ a*(SB^2-SC^2)*SA*(SA^2+5*S^2)* v*w ]=0 (trilinears) = isogonal conjugate of the line {4, 2889, 6101, 11591}. O is the only ETC center on it.

 

Zn = NaNbNc->ABC:  The locus is the line {3, 143, 1173, 3060, 3567, 5422, 5495, 5946, 9777, 10263}.  

If OP=t*OH, then OZn=t*OX(143).

 

Zn(O) = O ; Zn(H)=X(143)

  

Za(O) = isogonal conjugate of X(11591)

= 1/((b^2+c^2)*a^2-(b^2-c^2)^2)/ (a^4-2*(b^2+c^2)*a^2+3*b^2*c^ 2+c^4+b^4) : : trilinears

= on lines: {30,54}, {95,3260}, {1990,8882}

= isogonal conjugate of X(11591)

= trilinear pole of the line {1637,2623}

= [ -5.392942000760921, 11.54709120356563, -1.864425427902168 ]

 

Za(H) = isogonal conjugate of X(6101)

= 1/(a*((b^2+c^2)*a^6-(3*b^4+4* b^2*c^2+3*c^4)*a^4+(b^2+c^2)*( 3*b^4-b^2*c^2+3*c^4)*a^2-(b^6- c^6)*(b^2-c^2))) : : (trilinears)

= on lines: {5,1614}, {53,10312}, {54,3613}, {311,1078}

= [ -5.946442603069915, -8.88166434592441, 12.534020999733550 ]

 

César Lozada