- 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. - 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.

> - Dear friends:

I now see that this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".

Here is the version for two circles:

(A) and (B) are circles

The line AB intersect (B) at M and N

M' and N' are the inverses of M and N with respect to (A)

J is the inverse of A with respect to (B)

O is the inverse of J with respect to (A)

A' is the reflection of A on O

The locus points P such that the polar of P with respect to (A) is tangent to (B) is a conic with foci A and A' and diameter M'N'.

what is the description of the locus in the general case in terms of the two given conics?

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

Dear Francisco, it is the polar conic of one conic with respect to another, in other words, the dual of a conic is a conic. For further properties you can see Geometry of conics by A.V. Akopyan and A.A. Zaslavsky, pages 70-72.

> Dear friends:

>

> I now see that this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".

>

> Here is the version for two circles:

>

> (A) and (B) are circles

> The line AB intersect (B) at M and N

> M' and N' are the inverses of M and N with respect to (A)

> J is the inverse of A with respect to (B)

> O is the inverse of J with respect to (A)

> A' is the reflection of A on O

> The locus points P such that the polar of P with respect to (A) is tangent to (B) is a conic with foci A and A' and diameter M'N'.

>

> what is the description of the locus in the general case in terms of the two given conics?

>

> Thank you.

>