A conic section passing through the common tangent points of two conic sections

Question

Nice feature I just came up with

We have two conic sections, and we draw the common tangents to the two conic sections. The points of contact on the two conic sections lie on one conic section.

Is this feature already known?

How do we prove that anyway?

Answer 1

This conic (I'll call it $F$) is discussed in the classical projective geometry literature. I've seen it variously referred to as "Salmon's conic", "covariant conic" and "harmonic locus". It has the remarkable property that, for any point $P$ on $F$, the four tangents from $P$ to the two original conics form a harmonic set, i.e. their cross ratio is $-1$.

For a synthetic treatment, see Hatton, Projective Geometry, pg. 293, Article 133.

For an analytical treatment, see Salmon, Conic Sections, pg 306, Article 334 (see also Article 378, pg 344).

The above references also discuss the dual curve. I've seen applications of $F$ to topics such as conics in double contact, and Poncelet's porism.