HYACINTHOS 26949

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

(Oa), (Ob), (Oc) = the circumcircles of NBC, NCA, NAB, resp.

(O1), (O2), (O3) = the reflections of (O) in BC, CA, AB, resp. [ = circumcircles of HBC, HCA, HAB, resp.]

Ra, Rb, Rc = the radical axes of ((N),(Oa)), ((N),(Ob)), ((N),(Oc)), resp.

 

R1, R2, R3 = the radical axes of ((N),(O1)), ((N),(O2)), ((N),(O3)), resp.

AaBbCc = the triangle bounded by Ra, Rb, Rc

A1B2C3 = the triangle bounded by R1, R2, R3

1. AaBbCc, A1B2C3 are perspective.
Perspector (on the Euler line)?

 

2. AaBbCc, A1B2C3 are orthologic.
The orthologic center (AaBbCc, A1B2C3) is the N.
The other one (on the Euler line)?

 

 

[Angel Montesdeoca]:


*** AaBbCc, A1B2C3 are perspective.     Perspector : X(403), point on the Euler line, inverse-in-nine-point-circle of X(4).

**** The orthologic center ( A1B2C3, AaBbCc), lies on the Euler line and is the midpoint the X(4) and X(14865)

a^8 (b^2+c^2)
-2 a^6 (b^4+3 b^2 c^2+c^4)
+2 a^2 (b^2-c^2)^2 (b^4+3 b^2 c^2+c^4)
-(b^2-c^2)^4 (b^2+c^2) : ... : ...

Lies on lines X(i)X(j) for these {i, j}: {2,3}, {6,11457}, {11,9628}, {53,9606}, {54,1503}, {74,13568},{125,10110}, {232,9698}, {254,8801}, {264,1238}, {511,6152}, {578,11550}, {973,2781}, {1092,3818}, {1853,10982}, ....

 

Is the midpoint  X(4) and X(14865)

 

Is the reflection of X(i) in X(j), for these {i, j}: {13160,5576}, {13564,7568}.

 (6 - 9 - 13) - search numbers  of W: (-92.9092195293278, -93.5336425851799, 111.275902977491).

Angel Montesdeoca