HYACINTHOS 29561

[Antreas P. Hatzipolakis]

Let ABC be a triangle and L a line.

Denote:

A', B', C' = intersections of L and BC, CA, AB, resp.

Oa, Ob, Oc = the circumcenters of AB'C', BC'A', CA'B', resp.

For:
1. L = OI line:
The Feuerbach hyperbola center X(11) of OaObOc lies on the OI line.
Hyacinthos  29554  

 

2. L = OH line (Euler line)
The Jerabek hyperbola center X(125) of OaObOc lies on the OH line

.3. L = OK line (Brocard axis)
The Kiepert hyperbola center X(115) lies on the OK line.

Which are 2 and 3 points wrt trisngle ABC?

[Angel Montesdeoca]:

**** 2. L = OH line (Euler line)
The Jerabek hyperbola center X(125) of OaObOc is X(402) of ABC, lies on the OH line


**** 3. L = OK line (Brocard axis)

The Kiepert hyperbola center X(115) lies on the OK line.

 

 W = X(3)X(6) ∩ X(232)X(14356) =

 

= a^2 (-b^4-c^4+a^2 (b^2+c^2)) (a^6-a^4 (b^2+c^2)-(b^2-c^2)^2 (b^2+c^2)+a^2 (b^4-b^2 c^2+c^4)) : ;
 
= lies on these lines: {3,6}, {232,14356}, {620,2492}, {4045,18122}, {7669,22146}, {8574,21203} 
 
 Angel  Montesdeoca