HYACINTHOS 26973

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Ab, Ac = the orthogonal projections of A on IB, IC, resp.
Bc, Ba = the orthogonal projections of B on IC, IA, resp.

Ca, Cb = the orthogonal projections of C on IA, IB, resp.

La, Lb, Lc = the Euler lines of AAbAc, BBcBa, CCaCb, resp. concurrent at the Feuerbach point Fe.

Oa, Ob, Oc = the circumcenters of AAbAc, BBcBa, CCaCb, resp. [ =midpoints of AI, BI,CI, resp.]

The perpendiculars to La, Lb, Lc at Oa, Ob, Oc, resp. bound triangle A*B*C*.

The Euler line of A*B*C* passes through N, I, Fe of ABC.

Which are these points (N, I, Fe of ABC) wrt triangle A*B*C* ?

 

[The circumcenter O* of A*B*C* is the antipode of Fe in the NPC (N) of ABC]

 

 

[Peter Moses]:

Hi Antreas,

>Which are these points (N, I, Fe of ABC) wrt triangle A*B*C* ?

I = X(403).

N = a^2 (2 a^8-4 a^6 b^2+4 a^2 b^6-2 b^8-4 a^6 c^2+8 a^4 b^2 c^2-5 a^2 b^4 c^2+b^6 c^2-5 a^2 b^2 c^4+2 b^4 c^4+4 a^2 c^6+b^2 c^6-2 c^8):: 

= 3 X[2] + (2 J^2 - 3) X[3] 

on lines {{2,3},{35,10149},{74,10540},{155,1620},{156,1204},{567,15053},...}.
midpoint of X(i) and X(j) for these {i,j}: {{3,186},{74,10540},{548,10096},{550,11563},{2070,2071},{2072,10295},{5899,7464},{11558,12103}}.
reflection of X(i) in X(j) for these {i,j}: {{140,2072},{186,7575},{468,11563},{10096,11799},{10151,3627},{11695,13376},{12105,5899},{15350,546}}.
circumcircle inverse of X(382).
nine point circle inverse of X(10224).
psi-transform of X(9544).
X(382)-vertex conjugate of X(523).
reflection of X(11563) in the orthic axis.
...
...
Fe = X(186).


Bet regards,
Peter Moses.