HYACINTHOS 26506

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

A', B', C' = the reflections of P in BC, CA, AB, resp.

A", B", C" = the isogonal conjugates of P wrt triangles A'BC, B'CA, C'AB, resp.

 

For P = I:

 

1. ABC, A"B"C" are bilogic (perspective and orthologic).
Perspector = orthologic center (A"B"C", ABC) = H
Which is the other orthologic center (ABC, A"B"C") ?

2. ABC, A"B"C" are cyclologic.
Cyclologic centers?

Loci ??

[César Lozada]:

 

1)      Locus for perspectivity: a nasty degree-9 through ETC’s X(1)=I.

2)      Locus for orthology: {circumcircle}\/{A degree-6-curve, no ETC on it} \/ {a degree-7-curve through I,O}

For P=I, orthologic centers (54,  4)

For P=O, orthologic centers (Q , O), where

Q= X(4)X(93) ∩ X(50)X(252)
= (S^2-SA^2)*(3*S^2-SB^2)*(3*S^ 2-SC^2)*SB*SC : : (barycentrics)

= On lines: {4, 93}, {50, 252}, {54, 14106}, {930, 1299}, {2904, 8745}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4, 562, 93), (93, 562, 3519)

= [ -191.763897390410500, 63.18878431600996, 48.400997212551570 ]

 

3)      For P=I, cyclologic centers A<->A2 = (1157, 4)

 

César Lozada