HYACINTHOS 23148

Antreas Hatzipolakis

 

Let ABC be a triangle and P a pont.

Denote:

Pa,Pb,Pc  = the reflections of P in BC, CA, AB, resp.

Pab, Pac = the reflections of P in BPa, CPa, resp.
Pbc, Pba = the reflections of P in CPb, APb, resp.
Pca, Pcb = the reflections of P in APc, BPc, resp.

Which is the locus of P such that the perpendicular bisectors

of  PabPac, PbcPba, PcaPcb are concurrent?

And which is the locus of the point of concurrence?

APH

 

[César Lozada]:

 

Locus for concurrence=EulerLine (OH)U{A,B,C}

Locus of concurrence: Line (74,110)=Line(3,110)

If P=t*OH and Q=X(110)= Focus of Kiepert parabola, then

Z(P) = t*(t-1)/(t-(R/OH)^2)*OQ

 

Some ETC-pairs (P,Z(P)): (3,3), (4,3), (5,5944), (25,6090)

 

Other:

For P=G

  Z(P) = (3*a^4-2*(b^2+c^2)*a^2-b^4-c^4)*a : : (trilinears)

       = 2*cos(2*A)*cos(B-C)+5*cos(A)-3*cos(3*A) : : (trilinears)

       = On lines: (2,154), (3,74), (6,23) and more

       = ( -11.820395350978090, -15.62383815778222, 19.912734906977340 )       

 

Regards

César Lozada