## A conic centered at Euler line

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• 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
Message 1 of 4 , Oct 26, 2012
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
Message 2 of 4 , Oct 26, 2012
Sorry for the character codes,

See

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

--- 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
Message 3 of 4 , Oct 26, 2012
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.
• ... 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
Message 4 of 4 , Nov 4 10:14 PM
--- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
>
> 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.
>

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.
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