HYACINTHOS 28446

[Antreas P. Hatzipolakis]:

 

 

 

Let ABC be a triangle and L the Euler line.

Denote:

La, Lb, Lc = the reflections of L in BC, CA, AB, resp.

A', A" = the orthogonal projections of A on L, La, resp.

B', B" = the orthogonal projections of B on L, Lb, resp.

C', C" = the orthogonal projections of L on L, Lc, resp.

The circumcircles of AA'A", BB'B", CC'C" are coaxial.

Radical trace?

 

 

[Angel Montesdeoca]:

   
****    The radical trace of the circumcircles of AA'A", BB'B", CC'C":  

 

 =  MIDPOINT OF X(107) AND X(125)

Barycentrics (b^2-c^2)^6 (b^6+6 b^4 c^2+6 b^2 c^4+c^6)-(b^2-c^2)^4 (4 b^8-5 b^6 c^2-32 b^4 c^4-5 b^2 c^6+4 c^8) a^2-46 b^2 c^2 (b^2-c^2)^4 (b^2+c^2) a^4+(b^2-c^2)^2 (19 b^8+26 b^6 c^2-94 b^4 c^4+26 b^2 c^6+19 c^8) a^6-(b^2-c^2)^2 (27 b^6-38 b^4 c^2-38 b^2 c^4+27 c^6) a^8+(4 b^8-55 b^6 c^2+104 b^4 c^4-55 b^2 c^6+4 c^8) a^10+2 (7 b^6-9 b^4 c^2-9 b^2 c^4+7 c^6) a^12+(-5 b^4+22 b^2 c^2-5 c^4) a^14-4 (b^2+c^2) a^16+2 a^18  : :

= lies on these lines: {4,74}, {122,6723}, {402,5972}, {10745,23515}, {12295,23240}, {16111,22337}, {16163,23239}.  

=  midpoint of X(i) and X(j), for these {i, j}: {107,125}, {12295,23240}, {16111,22337}.

= reflection of X(i) in X(j), for these {i, j}: {122,6723}, {5972,6716}.

 (6 - 9 - 13) - search numbers  of W: (1.86781484480897, 1.37869550233370, 1.82411459037998).

Angel Montesdeoca