HYACINTHOS 26234

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the reflections of I in Na,Nb, Nc, resp.

Ni, Nii, Niii = the reflections of Na, Nb, Nc in I resp.

1. The circumcircles of ON1Ni, ON2Nii, ON3Niii are coaxial.

2. The NPC centers of ON1Ni, ON2Nii, ON3Niii are collinear.

[César Lozada]:

 

1)      2nd point of intersection (other than O)

Q2 = reflection of X(6789) in X(13607)

= 4*p^5*(4*p-17*q)+4*(17*q^2+12) *p^4-(28*q^2+73)*q*p^3+(4*q^4+ 52*q^2-9)*p^2-(17*q^2-6)*q*p+( 2*q^2-1)*q^2 : : (trilinears), where p=sin(A/2), q=cos((B-C)/2)

= On lines: {3,519}, {952,11717}, {3244,3667}, {6789,13607}

= reflection of X(6789) in X(13607)

= [ 1.986941081214689, 5.57525613919929, -1.136177959637295 ]

 

2)      On the line {140, 900}

 

César Lozada