HYACINTHOS 25952

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.
A2, A3 = the reflections of B', C' in Ab, Ac, resp.

Bc, Ba = the orthogonal projections of B' on BA, BC, resp.

B3, B1 = the reflections of C', A' in Bc, Ba, resp.

 

Ca, Cb = the orthogonal projections of C' on CB, CA, resp.

C1, C2 = the reflections of A', B' in Ca, Cb, resp.

 

[(Oa), (Ob), (Oc)], [(Na), (Nb), (Nc)] = the circumcircles, NPCs, resp. of AA2A3, BB3B1, CC1C2, resp.

1. P = I

(Oa), (Ob), (Oc) concur at the reflection of I in Feuerbach point = X80

2. P = O

(Na), (Nb), (Nc) concur. 

Point?

Note: It is similar construction with the point X1986

 

[Peter Moses]:

 

Hi Antreas,

 

2).

a^2 (3 a^8 b^2-6 a^6 b^4+6 a^2 b^8-3 b^10+3 a^8 c^2+10 a^6 b^2 c^2-7 a^4 b^4 c^2-9 a^2 b^6 c^2+3 b^8 c^2-6 a^6 c^4-7 a^4 b^2 c^4+16 a^2 b^4 c^4-9 a^2 b^2 c^6+6 a^2 c^8+3 b^2 c^8-3 c^10):: 

on lines {{23, 110}, {389, 12308}, {5663, 12811}, {6723, 10219}, {9729, 10620}}.

8 X[6723] - 9 X[10219], 3 X[9729] - X[10620], 3 X[389] + X[12308].

 

Best regards,

Peter Moses.