HYACINTHOS 27242

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point, A'B'C' the pedal triangle of P and (N) = the common pedal circle of H and O [ = the NPC].

Denote:

(Npa), (Npb), (Npc) = the reflections of (N) in PA', PB', PC', resp.

Ra = the radical axis of (O), (Npa)
Rb = the radical axis of (O), (Npb)
Rc = the radical axis of (O), (Npc)

A*B*C* = the triangle bounded by Ra, Rb, Rc.

3. P = midpoint of ON = X(140):
3.1. ABC, A*B*C* are homothetic. Homothetic center ?
3.2. ABC, A*B*C* are orthologic.
Orthologic center (A*B*C*, ABC) = Orthocenter of A*B*C* ?

[César Lozada]:

 

3.1)

Q = X(74)X(140) ∩ X(98)X(468)

= (SB+3*SC)*(SC+3*SB)*(S^2-3*SA* SB)*(S^2-3*SA*SC) : : (barys)

= on lines: {74, 140}, {98, 468}, {5627, 10096}, {6676, 14919}

= [ 0.4830953767701150, -0.7194118802237510, 3.9157517635530780 ]

 

3.2)

A*->A’: X(4)

A’->A* = X(140)X(2777) ∩ X(523)X(6140)

= 3*S^4+(504*R^4+6*R^2*(2*SA-41* SW)-SB*SC+27*SW^2)*S^2+3*(72* R^4+2*R^2*SW-3*SW^2)*SB*SC : : (barys)

= on lines: {140, 2777}, {523, 6140}

= [ -1.1281776832231330, -2.2396640960019170, 5.7118985560117660 ]

 

César Lozada