HYACINTHOS 25121

[Antreas P. Hatzipolakis]:

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

Denote

Ha, Hb, Hc = the reflections of H in B'C', C'A', A'B', resp.

Haa, Hbb, Hcc = the reflections of Ha,Hb,Hc in BC, CA, AB, resp.

H1, H2, H3 = the reflections of H in BC, CA, AB, resp.

H11, H22, H33 = the reflections of H1, H2, H3 in B'C', C'A', A'B', resp.

A*B*C* = the triangle bounded by HaaH11, HbbH22, HccH33

1. A'B'C', HaaHbbHcc are homothetic.
Homothetic center (on the Euler line) ?

2. ABC, A*B*C* are parallelogic.

The parallelogic center (ABC, A*B*C*) is the G.

The other one?

EXCENTRAL TRIANGLE VERSION

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

Denote

Ia, Ib, Ic = the reflections of I in BC,CA, AB, resp.

Iaa, Ibb, Icc = the reflections of Ia,Ib,Ic in B'C', C'A', A'B', resp.

I1, I2, I3 = the reflections of I in B'C', C'A', A'B', resp.

I11, I22, I33 = the reflections of I1, I2, I3 in BC, CA, AB, resp.

A*B*C* = the triangle bounded by IaaI11, IbbI22, IccI33

3. ABC, IaaIbbIcc are homothetic.
Homothetic center (on the Euler line of A'B'C') ?

4. A'B'C', A*B*C* are parallelogic.

The parallelogic center (A'B'C', A*B*C*) is the G of A'B'C'.

The other one?

[Angel Montesdeoca]:

ORTHIC TRIANGLE VERSION

1. A'B'C', HaaHbbHcc are homothetic.
Homothetic center (on the Euler line) ?

****   X(1594) = Rigby-Lalescu orthopole

2. ABC, A*B*C* are parallelogic.
The parallelogic center (ABC, A*B*C*) is the G.
The other one?

****  V =  X(3)X(1199) /\ X(858)X(6243)

V = (a^2 ((b^2-c^2)^4 (2 b^6+3 b^4 c^2+3 b^2 c^4+2 c^6)
     -2 (b^2-c^2)^2 (4 b^8+4 b^6 c^2+3 b^4 c^4+4 b^2 c^6+4 c^8) a^2
     +2 (5 b^10+b^8 c^2-2 b^6 c^4-2 b^4 c^6+b^2 c^8+5 c^10) a^4
     -2 b^2 c^2 (b^4-b^2 c^2+c^4) a^6
     +(-10 b^6-11 b^4 c^2-11 b^2 c^4-10 c^6) a^8
     +2 (4 b^4+5 b^2 c^2+4 c^4) a^10
     -2 (b^2+c^2) a^12) : ... : ... ),
    
    with (6-9-13)-search numbers ( 9.13527852082385, 7.29272565695836, -5.62442798252088).
   
EXCENTRAL TRIANGLE VERSION

    3. ABC, IaaIbbIcc are homothetic.
Homothetic center (on the Euler line of A'B'C') ?


**** X(165) on the Euler line of A'B'C':  X(1)X(3)

4. A'B'C', A*B*C* are parallelogic.
The parallelogic center (A'B'C', A*B*C*) is the G of A'B'C'.
The other one?

**** W = X(35)X(1699) /\ X(40)X(382)

W = (2 (b-c)^4 (b+c)^3
    -(b-c)^2 (b+c)^4 a
    -(b-c)^2 (b^3+c^3) a^2
    -(b-c)^2 (2 b^2+3 b c+2 c^2) a^3
    +(-4 b^3-3 b^2 c-3 b c^2-4 c^3) a^4
    +(7 b^2+b c+7 c^2) a^5
    +3 (b+c) a^6
    -4 a^7 : ... : ...),
   
     with (6-9-13)-search numbers (-12.6921889638726, -13.6867545764541, 18.9740433258553).
    
     Angel Montesdeoca