HYACINTHOS 26152

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the cevian triangle of G.

Denote:

(Oa) = the reflection of the circle with diameter AA' in BC.

(Ob) = the reflection of the circle with diameter BB' in CA.

(Oc) = the reflection of the circle with diameter CC' in AB.

 

Which point is the radical center of (Oa), (Ob), (Oc) (on the Jerabek hyperbola) ?

 

 

[Peter Moses]:

Hi Antreas,

 

a^2 (2 a^4-4 a^2 b^2+2 b^4-3 a^2 c^2-3 b^2 c^2+c^4) (2 a^4-3 a^2 b^2+b^4-4 a^2 c^2-3 b^2 c^2+2 c^4):: 

 

on lines {{2,1493},{3,1199},{4,11423},{ 6,3518},{54,11202},{64,7592},{ 68,3090},{69,575},{70,11427},{ 74,578},{184,1173},{248,7772}, {265,3091},{323,13154},{389, 3431},{568,12226},{576,1176},{ 879,7950},{3146,3521},{3147, 5486},{3426,11426},{3520,3532} ,{3527,10594},{3529,4846},{ 3531,5198},{3628,9716},{5890, 11270},{6241,11738},{6391, 9925},{6413,6420},{6414,6419}, {6415,6428},{6416,6427},{ 10261,10783},{10262,10784},{ 11004,13353}}.

on Jerabek.

isogonal of X(1656).

X(i)-cross conjugate of X(j) for these (i,j): {{3567, 4}, {13351, 2}}.

isoconjugate of X(j) and X(j) for these (i,j): {{1, 1656}, {92, 10979}}.

cevapoint of X(i) and X(j) for these (i,j): {{6, 11402}, {184, 13345}}.

trilinear pole of line {647, 1510}.

barycentric quotient X(i)/X(j) for these {i,j}: {{6, 1656}, {184, 10979}, {8882, 4994}}.

Searches: {1. 38045262479048546632760615716, 2. 68588827098646512011226977556, 1. 14407139055198688165865856996} .

 

 

Best regards,

Peter Moses.