HYACINTHOS 23098

[Seiichi Kirikami]:

 

Dear friends,

 

Let ABCDEF be a hexagon and AD, BE, CF concur in a point P.

Denote:

Oa=the circumcenter of APF,

Ob=the circumcenter of CPB,

Oc=the circumcenter of EPD,

O1=the circumcenter of BPA,

O2=the circumcenter of DPC,

O3=the circumcenter of FPE,

1) OaObOc and O1O2O3 are cyclologic at points Q and R.

2) Oa, Ob, Oc, O1, O2, O3 are on a conic K.

3) Q and R are on K.

See the attached picture.

 

A synthetic proof is needed because algebraic computation often fails in case of cyclology.

 

Best regards,

Seiichi.

 

[APH]:

Dear Seiichi,

If we take OaObOc as reference trianlgle ABC then it is equivalent to:

Let ABC be a triangle P a point, and L1, L2, L3 three lines passing through P.

Denote:
C2 = the other than P intersection of the line L2 and the circle (C, CP)
B3 = the other than P intersection of the line L3 and the circle (B, BP)
O1 = the circumcenter of PC2B3
Similarly O2, O3
The triangles ABC, O1O2O3 are cycllologic.

We can apply it in different ways in order to get triangle centers

Here is such a way:

Let P be the Schiffler point and L1, L2, L3 the Euler lines of IBC, ICA, IAB (cncurrent at P)

(And in general let Q be a point on the Neuberg cubic and P the intersection
of the Eluler lines L1, L2, L3 of QBC, QCA, QAB, resp)

 

Antreas P, Hatzipolakis

************************************************

[Angel Montesdeoca]:

[APH]: 

 

Let P=X(21) be the Schiffler point and L1, L2, L3 the Euler lines of IBC, ICA, IAB (concurrent at P)

Denote:

C2 = the other than P intersection of the line L2 and the circle (C, CP)

B3 = the other than P intersection of the line L3 and the circle (B, BP)

O1 = the circumcenter of PC2B3

Similarly O2, O3

The triangles ABC, O1O2O3 are cycllologic.

 

*** The first barycentric coordinate of the  cyclologic center  ABC-O1O2O3 is:

a (a^9+a^7 (-6 b^2+7 b c-6 c^2)-(b-c)^4 (b+c)^3 (2 b^2+5 b c+2 c^2)+a^6 (2 b^3-3 b^2 c-3 b c^2+2 c^3)+a (b^2-c^2)^2 (3 b^4-b^3 c+2 b^2 c^2-b c^3+3 c^4)+a^5 (12 b^4-15 b^3 c+13 b^2 c^2-15 b c^3+12 c^4)+a^4 (-6 b^5+3 b^4 c+5 b^3 c^2+5 b^2 c^3+3 b c^4-6 c^5)+a^3 (-10 b^6+9 b^5 c-3 b^4 c^2+5 b^3 c^3-3 b^2 c^4+9 b c^5-10 c^6)+a^2 (6 b^7+3 b^6 c-11 b^5 c^2+b^4 c^3+b^3 c^4-11 b^2 c^5+3 b c^6+6 c^7))

 

Exact trilinear coordinates  at the tirángulo ETC (6-9-13):

(3.10306953362835054,4.70121811205322202,-1.04621091888093639)