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.