HYACINTHOS 21297

Dear Angel,

If you substitute ellipses for hyperbolas in this construction, the resulting polars are not concurrent, but form a triangle which is perspective to ABC at ETC center X(4320).

Best regards,
Randy Hutson

 

--- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@...> wrote:
>
> Dear Hyacinthists.
>
> (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143)
>
> Given triangle ABC, consider the hyperbola passing through A, and with foci at B and C. We shall call this the A-Soddy hyperbola of the triangle.
>
> The equation of A-Soddy hyperbola is
> (Fa): (c+a-b)(a+b-c)(y^2+z^2)-2(a^2+(b-c)^2)y z+4(b-c)c x y -4b(b-c)z x=0.
>
> The perspector of a (Fa): Pa = ( SB SC : - b c SC : -b c SB ).
>
> The polar of Pa with respect to (Fa) is is the line "da":
> 2b(b-c)cx + (a^2+(b-c)^2)cy - b(a^2+(b-c)^2)z=0.
>
> Similarly define the lines "db" and "dc"; then the lines da, db and dc are concurrent at the triangle center X(4319).
>
>
>
> Sincerely
> Angel Montesdeoca