HYACINTHOS 26069

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

A1, A2 = orthogonal projections of N on AH, AO, resp.
B1, B2 = orthogonal projections of N on BH, BO, resp.
C1, C2 = orthogonal projections of N on CH, CO, resp.

M1, M2, M3 = the midpoints of A1A2, B1B2, C1C2, resp.

 

N, M1, M2, M3 are concyclic.

Center of the circle ?

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ABC, M1M2M3 are homothetic.

Homothetic center?

 

[Peter Moses]:

Hi Antreas,

>Center of the circle ?

X(12006)

>ABC, M1M2M3 are homothetic.

at
a^2 (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2-3 b^6 c^2+3 a^4 c^4+3 a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-3 b^2 c^6):: 

on lines
{{1,59},{2,578},{3,143},{4, 569},{5,49},{6,5889},{20,182}, {22,10982},{23,10110},{24, 5640},{26,9781},{30,13353},{ 51,7488},{52,7691},{60,2617},{ 97,2055},...}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,569,5012),(5,54,110),(5, 567,54),(6,7503,5889),(54,110, 9706),(182,11424,20) ...
X(i)-aleph conjugate of X(j) for these (i,j): {{21, 2940}, {6727, 1048}}.
barycentric product X(249)X(8902).
barycentric quotient X(8902)/X(338).

Best regards,
Peter Moses.