Sorry for the character codes,

See

http://garciacapitan.blogspot.com.es/2012/10/a-conic-centered-at-euler-line.html
for a more readable version

--- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:

>

> I found the following:

>

> Theorem. Given a triangle ABC, call Γ the locus of points P such that polar of P with respect to the circumcircle is tangent to the nine point circle. Then we have:

> 1) Γ is a conic whose center is X26, the circumcenter of the tangential triangle.

> 2) Γ is an ellipse, parabola o hyperbola if and only if the triangle is acute, triangle or obtuse.

> 3) The diameter on the transverse axis of the conic is also a diameter of the circumcircle of tangential triangle. Therefore, the transverse axis of the conic is the Euler line of the triangle.

> 4) If α, β are the lengths of the transverse and conjugate axes of the conic respectively, we have the relation

>

> (αβ)^2=1−OH^2/R^2

>

> 5) The foci of the conic are O and O′, where O′ is the reflection of O on the center X26.

>