HYACINTHOS 24855

[Antreas P. Hatzipolakis]:

 

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

Denote:

Oa, Ob, Oc = the reflections of O on BC,CA, AB, resp.

Oaa, Oab, Oac = the orthogonal projections of Oa on NA', NB', NC', resp.
Oba, Obb, Obc = the orthogonal projections of Ob on NA', NB', NC', resp.
Oca, Ocb, Occ = the orthogonal projections of Oc on NA', NB', NC', resp.

We have:  Oba = Oca =: A", Ocb = Oab = : B", Oac = Obc =: C"

 

Na, Nb, Nc = the NPC centers of OaaB"C", A"ObbC", A"B"Occ, resp.

A'B'C', NaNbNc are congruent and homothetic.

Homothetic center (on the line at infinity)?

A"B"C", NaNbNc are inversely homothetic.

Homothetic center?

[Peter Moses]

Hi Antreas,

 

>A'B'C', NaNbNc are congruent and homothetic.

>Homothetic center (on the line at

infinity)?
X(5663).

 

>A"B"C", NaNbNc are inversely homothetic.

a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+2 a^4 b^2 c^2+6 a^2 b^4 c^2-9 b^6 c^2-3 a^4 c^4+6 a^2 b^2 c^4+20 b^4 c^4+3 a^2 c^6-9 b^2 c^6-c^8)::
on lines {{5,113},{143,3091},{381,10263 },{511,3856},{546,1216},...}.

(J^2 - 8) X[5] - J^2 X[113],  J = |OH|/R.

9 X[5] - X[185], 17 X[5] - 9 X[373], 3 X[546] + X[1216], X[143] - 5 X[3091], 11 X[5] - 3 X[9730], 9 X[381] - X[10263].

 

Best regards,

Peter Moses.