HYACINTHOS 28449

[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?

 

 

[Peter Moses]:

Hi Antreas,

Radical trace:  =  MIDPOINT OF X(107) AND X(125)

Barycentrics 2 a^18-4 a^16 b^2-5 a^14 b^4+14 a^12 b^6+4 a^10 b^8-27 a^8 b^10+19 a^6 b^12-4 a^2 b^16+b^18-4 a^16 c^2+22 a^14 b^2 c^2-18 a^12 b^4 c^2-55 a^10 b^6 c^2+92 a^8 b^8 c^2-12 a^6 b^10 c^2-46 a^4 b^12 c^2+21 a^2 b^14 c^2-5 a^14 c^4-18 a^12 b^2 c^4+104 a^10 b^4 c^4-65 a^8 b^6 c^4-127 a^6 b^8 c^4+138 a^4 b^10 c^4-12 a^2 b^12 c^4-15 b^14 c^4+14 a^12 c^6-55 a^10 b^2 c^6-65 a^8 b^4 c^6+240 a^6 b^6 c^6-92 a^4 b^8 c^6-77 a^2 b^10 c^6+35 b^12 c^6+4 a^10 c^8+92 a^8 b^2 c^8-127 a^6 b^4 c^8-92 a^4 b^6 c^8+144 a^2 b^8 c^8-21 b^10 c^8-27 a^8 c^10-12 a^6 b^2 c^10+138 a^4 b^4 c^10-77 a^2 b^6 c^10-21 b^8 c^10+19 a^6 c^12-46 a^4 b^2 c^12-12 a^2 b^4 c^12+35 b^6 c^12+21 a^2 b^2 c^14-15 b^4 c^14-4 a^2 c^16+c^18 ::

=X[5667] + 3 X[14644], X[107] - 3 X[14847], X[125] + 3 X[14847], X[16163] - 3 X[23239], X[10745] - 3 X[23515].
= on the cubic K818 and 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}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 14847, 107).


Best regards,
Peter Moses.