HYACINTHOS 25945

[Antreas P. Hatzipolakis]:

 

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

Denote:

Ba, Ca = the orthogonal projections of B', C' on BA, CA, resp.

Cb, Ab = the orthogonal projections of C', A' on CB, AB, resp.

Ac, Bc = the orthogonal projections of A', B' on AC, BC, resp.

 

 

A" = AA' Intersection BaCa
B" = BB' Intersection CbAb
C" = CC' Intersection AcBc


A* = BC Intersection B"C"
B* = CA Intersection C"A"
C* = AB Intersection A"B"

A*, B*, C* are coillinear.
(perspectrix of ABC, A"B"C")

 

Which point is the intersection of the line A*B*C* and the trilinear polar of I (antiorthic axis)?

[Peter Moses]:

Hi Antreas,

 

>Which point is the intersection of the line A*B*C* and the trilinear polar of I (antiorthic axis)?

 

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

on lines {{44,513},{521,4976},{2488, 11934},{3738,4765},{4131,4762} ,{8774,11068}}.

reflection of X(11934) in X(2488).

{X(650),X(4790)}-harmonic conjugate of X(654).

X(42)-complementary conjugate of X(5522).

crosspoint of X(651) and X(3296).

crossdifference of every pair of points on line {1, 6883}.

crosssum of X(650) and X(3295).

barycentric product X(i)X(j) for these {i,j}: {{513, 10527}, {522, 3338}, {3737, 12609}}.

barycentric quotient X(i)/X(j) for these {i,j}: {{663, 7162}, {3338, 664}, {10527, 668}}.

 

Best regards,

Peter Moses.