HYACINTHOS 20181

Dear Randy,

1. The ellipse you found also can be constructed as the ellipse through the traces of X(1) and X(2).
2. The ellipse you found also can be constructed as the center of each conic through A,B,C,X(1). (5th point can be any point unequal A,B,C)
3. I found these points to lie on this ellipse:
Midpoints of the sides of the Reference Triangle,
X(11), X(214), X(244), X(1015).

Best regards,

Chris van Tienhoven

 

--- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
>
> Friends,
>
> I would think this has been covered before, but I am unable to find anything on it:
>
> The locus of the centers of all circumhyperbolas passing through X(1) and a point on the Euler line is an ellipse with center X(1125). It passes through X(11) (center of Feuerbach Hyperbola) and X(1015) (Exsimilicenter of Moses Circle and Incircle). Are there any other Kimberling centers on this ellipse? I have checked the centers of circumhyperbolas passing through X(1) and Euler line points up to X(297), and have not found any. X(1015) is the center of hyperbola {A,B,C,X(1),X(2)}. The hyperbola {A,B,C,X(1),X(3)} also passes through X(29), and its center is not a Kimberling center.
>
> Is anything else known about this ellipse?
>
> Regards,
> B. Randy Hutson
>