HYACINTHOS 27039

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the cevian triangle of Η.

Denote:

Ab, Ac = the orthogonal projections of A' on BB', CC', resp.
A2, A3 = the orthogonal projections of Ab, Ac on BC, resp.

Bc, Ba = the orthogonal projections of B' on CC', AA', resp.
B3, B1 = the orthogonal projections of Bc, Ba on CA, resp.

 

Ca, Cb = the orthogonal projections of C' on AA', BB', resp.
C1, C2 = the orthogonal projections of Ca, Cb on AB, resp.

M1, M2, M3 = the midpoints of B1C1, C2A2, A3B3, resp.

A*B*C* = the triangle bounded by B1C1, C2A2, A3B3

 

1. ABC, M1M2M3 are perspective.
2. A'B'C', A*B*C* are perspective.

 

 

[Peter Moses]:

Hi Antreas,

1). X(32).

2).
a^4 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2-b^4+a^2 c^2-b^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4):: 

on the lines {{32,682},{211,232},{389,1595},{2387,8743},{14569,14715}}.
on the Kiepert hyperbola of the orthic triangle.
orthic isogonal conjugate of X(3199).
X(4)-Ceva conjugate of X(3199).
barycentric product X(i)X(j) for these {i,j}: {{53, 160}, {324, 3202}, {2979, 3199}}.
barycentric quotient X(3202)/X(97).

Best regards,
Peter Moses.