HYACINTHOS 25977

[Antreas P. Hatzipolakis]:

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

Denote:

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

 

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

 

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

C1, C2 = the orthogonal projections of Ca, Cb on CC', resp.

A*B*C* = the triangle bounded by B3C2, C1A3, A2B1

A'B'C', A*B*C* are homothetic.

Homothetic center  in terms of P?

[Peter Moses]:

 

Hi Antreas,

 

p ((a^2+b^2-c^2) p q r^2+q^2 ((a^2-b^2+c^2) p r+2 a^2 r^2)+p^2 (c^2 q^2+b^2 r^2))::

 

X(1) –> X(354).

X(2) –> X(597).

X(4) –> X(427).

 

X(3) –> 

a^2 (a^2-b^2-c^2) (a^4 b^4-2 a^2 b^6+b^8+2 a^2 b^4 c^2-2 b^6 c^2+a^4 c^4+2 a^2 b^2 c^4+2 b^4 c^4-2 a^2 c^6-2 b^2 c^6+c^8):: 

on lines {{2,1972},{3,54},{5,6662},{6,4 26},{51,852},{216,3289},{264,1 1197},{389,417},{394,6641},{40 8,970},{418,511},...}

{X(51),X(6509)}-harmonic conjugate of X(852).

X(9251)-anticomplementary conjugate of X(2888).

crosspoint of X(3) and X(264).

crosssum of X(4) and X(184).

barycentric product X(i)X(j) for these {i,j}: {{76,6752},{394,6747}}.

 

X(6) –> 

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

on lines {{6,110},{23,5038},{39,51},{18 2,8627},{251,12834},{373,3231} ,...}.

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{6,5640,3124},{373,5052,3231} }.

crosspoint of X(6) and X(598).

crosssum of X(2) and X(574).

 

Best regards,

Peter Moses.