HYACINTHOS 29476

[Antreas P. Hatzipoklakis]:

Let ABC be a triangle, HaHbHc the cevian triangle of H (orthic triangle), GaGbGc the cevian triangle of G (medial triangle), H1H2H3, G1G2G3 the circumcevian triangles of H and G, resp.

Denote:

A1 = the 2nd intersection of H1Ga and the circumcircle (other than H1)
A2 = the 2nd intersection of G1Ha and the circumcircle (other than G1)

B1 = the 2nd intersection of H2Gb and the circumcircle (other than H2)
B2 = the 2nd intersection of G2Hb and the circumcircle (other than G2)

C1 = the 2nd intersection of H3Gc and the circumcircle (other than H3)
C2 = the 2nd intersection of G3Hc and the circumcircle (other than G3)

The lines A1A2, B1B2, C1C2 bound a triangle A*B*C*
(A* = B1B2 /\ C1C2, B* = C1C2 /\ A1A2, C* = A1A2 /\ B1B2)

The triangles ABC, A*B*C* are perspective.

[Francisco Javier Garcia Capitan]:

The coordinates are:
 
a^2 (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (3 a^4 - 6 a^2 b^2 + 3 b^4 + c^4) (3 a^4 - 6 a^2 c^2  +  b^4  + 3 c^4) : :
not in ETC

[Antreas P. Hatzipolakis]:

Is this point now in ETC ?

[Peter Moses]:

 

Hi Antreas,

>Is this point now in ETC ?

No.

ISOGONAL CONJUGATE OF X(23291) =

= a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^4 - 6*a^2*b^2 + 3*b^4 + c^4)*(3*a^4 + b^4 - 6*a^2*c^2 + 3*c^4) : :

= lies on these lines: {20, 6530}, {232, 9909}, {325, 6353}, {511, 3515}, {523, 30549}, {9306, 15394}, {14575, 22263}

= isogonal conjugate of X(23291)
= cevapoint of X(i) and X(j) for these (i,j): {3, 8780}, {25, 8778}, {154, 3053}, {577, 1974}
= barycentric quotient X (6)/X(23291)

Best regards,
Peter Moses.