HYACINTHOS 25073

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A1B1C1 the pedal triangle of H.

Denote:

A2, B2, C2 = the reflections of H in BC, CA, AB, resp.

(Oab), (Oac) = the circumcircles of AB1C2, AB2C1, resp.

(Obc), (Oba) = the circumcircles of BC1A2, BC2A1, resp.

(Oca), (Ocb) = the circumcircles of CA1B2, CA2B1, resp.

R1 = the radical axis of (Oab), (Oac)

R2 = the radical axis of (Obc), (Oba)

R3 = the radical axis of (Oca), (Ocb)

R1, R2, R3 are concurrent.

 

 

[César Lozada]:

In general, these radical axis concur for any P on the Neuberg cubic.

 

If P=u:v:w (trilinears) and P is on the Neuberg cubic, then the point of concurrence Z(P) is:

Z(P) = u^2 :  v^2*(2*cos(C)*u+v)/(u+2*cos( C)*v) :  w^2*(2*u*cos(B)+w)/(u+2*cos( B)*w)

 

ETC-pairs (P,Z(P)): (1,1), (3,5562), (4,8884)

 

Some others:
Z(X(13)) = (sqrt(3)*(S^2+SB*SC) -S*(SA+6*R^2-3*SW))/(sqrt(3)* SA+S) : :  (barycentrics)

= (1-2*cos(2*A)+4*sin(A+Pi/6)* cos(B-C))*csc(A+Pi/3) ::  (trilinears)

= On cubic K060 and these lines: {5,8929}, {13,15}, {14,8014}, {61,11555}, {265,11139}, {618,11119}, {623,11078}, {1117,11071}, {1141,5995}, {6104,6671}

= [ 0.002589646566028, 0.00297440016044, 3.637410060304367 ]

 

Z(X(14)) = (sqrt(3)*(S^2+SB*SC) +S*(SA+6*R^2-3*SW))/(sqrt(3)* SA-S) : :  (barycentrics)

= (1-2*cos(2*A)-4*sin(A-Pi/6)* cos(B-C))*csc(A-Pi/3)::  (trilinears)

= On cubic K060 and these lines: {5,8930}, {13,8015}, {14,16}, {62,11556}, {265,11138}, {619,11120}, {624,11092}, {1117,11071}, {5479,5619}, {6105,6672}

= [ -0.614001735254890, 1.72515770371903, 2.729709949450383 ]

 

César Lozada