HYACINTHOS 25002

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and HaHbHc,OaObOc the pedal triangles of H,O, resp.

Denote:

A', B', C' = the reflections of N in BC, CA, AB, resp.

A1, B1, C1 = the orthogonal projections of A', B', C' on ObOc, OcOa, OaOb, resp.

A2, B2, C2 = the orthogonal projections of A', B', C' on HbHc, HcHa, HaHb, resp.

The circumcenters of A'A1A2, B'B1B2, C'C1C2 are collinear.

[Peter Moses]:

Hi Antreas,

 

>The circumcenters of A'A1A2, B'B1B2, C'C1C2 are collinear.

On the line through {512,5462,10279}, perpendicular to Brocard.

Intersection with Brocard axis is:

a^2 (a^10 b^2-5 a^8 b^4+10 a^6 b^6-10 a^4 b^8+5 a^2 b^10-b^12+a^10 c^2-5 a^6 b^4 c^2+12 a^4 b^6 c^2-12 a^2 b^8 c^2+4 b^10 c^2-5 a^8 c^4-5 a^6 b^2 c^4-2 a^4 b^4 c^4+5 a^2 b^6 c^4-7 b^8 c^4+10 a^6 c^6+12 a^4 b^2 c^6+5 a^2 b^4 c^6+8 b^6 c^6-10 a^4 c^8-12 a^2 b^2 c^8-7 b^4 c^8+5 a^2 c^10+4 b^2 c^10-c^12)::
on lines {{3,6},{249,1199},{512,5462},. ..} = X[52]+3 X[3111]

 

Best regards,

Peter Moses.