HYACINTHOS 25342

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C', A"B"C" the cevian triangles of H,G, resp.

Denote:

A* = the reflections of A" in B'C'

B* = the reflections of B" in C'A'

C* = the reflections of C" in A'B'

Ab, Ac = the orthogonal projections of A* on AC, AB, resp.

Bc, Ba = the orthogonal projections of B* on BA, BC, resp.

Ca, Cb = the orthogonal projections of C* on CB, CA, resp.

 Oa, Ob, Oc = the circumcenters of A*AbAc, B*BcBa, C*CaCb, resp.

1. A'B'C' are orthologic.

The orthologic center (OaObOc, A'B'C') is the midpoint of ON = X140

2. ABC, OaObOc are orthologic.

 

[Peter Moses]:

 

Hi Antreas,

 

1).

{OaObOc,A’B’C’} orthologic at X(140).

{A’B’C’, OaObOc} orthologic at:
a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-3 b^2 c^2+c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-6 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::
on lines {{4,2889},{6,1173},{23,9827},{ 185,7576},{428,6152},{1598, 2904},{1986,6756},{2914,5609}, {5895,11455},{7999,10516}}.

Searches {-2. 51951022758613447812683246361, -3. 79072232298120963846265690728, 7. 42786157978035553503565640613} .

5 X[1173]-7 X[9781].

X(5557) of the orthic triangle.

on the Feuerbach of the orthic triangle.

 

2)

{ABC,OaObOc} orthologic at X(1173).

{OaObOc,ABC} orthologic at X(5446).

 

Best regards,

Peter Moses.