HYACINTHOS 25964

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

Ba, Ca = the orthogonal projections of B', C' on BC, resp.
Mab, Mac = the midpoints of B'Ba, C'Ca, resp.

Cb, Ab = the orthogonal projections of C', A' on CA, resp.
Mbc, Mba = the midpoints of C'Cb, A'Ab, resp.

Ac, Bc = the orthogonal projections of A', B' on AB, resp.
Mca, Mcb = the midpoints of A'Ac, B'Bc, resp.

A*B*C* = the triangle bounded by MabMac, MbcMba, McaMcb.

ABC, A*B*C* are perspective.

1. Perspector?
2. Intersection point of the trilinear polar of I and the perspectrix
of (ABC, A*B*C*) ?

 

[Peter Moses]:

 

Hi Antreas,

1) a^2 (a^3-3 a b^2+2 b^3-a^2 c-3 b^2 c-a c^2+c^3) (a^3-a^2 b-a b^2+b^3-3 a c^2-3 b c^2+2 c^3):: 

on lines {{1,5785},{34,1419},{56,991},{ 86,11019},...}.
on ABCIK.
X(2488)-cross conjugate of X(101).
cevapoint of X(6) and X(2293).

isogonal conjugate of a simple point:

 2 a^3-3 a^2 b+b^3-3 a^2 c-b^2 c-b c^2+c^3:: 

on lines {{1,2},...,{7,165},...}.

2) X(650).

Best regards,
Peter Moses.