--- 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.