First time, I found a line associated with antipodal points, detail:

> Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ and parallel to $PA$, $PB$, $PC$ meets $BC$, $CA$, $AB$ respectively at three collinear points, the new line through the center of circumconic.

> **Question:** Is a line associated with antipodal points above known?

[![enter image description here][1]][1]

**Update**: But the fact, the result is generalization of the Simson line, I We reformulate as follows:

> Let $ABC$ be a triangle, $P$ be a point in the plane, let $C$ is the Nine point conic of $A$, $B$, $C$, $P$. Let $O$ be arbitrary point on $C$, $P'$ is the reflection of $P$ in $O$. Then three lines through $P'$ and parallel to $PA$, $PB$, $PC$ meet three lines $BC$, $CA$, $AB$ respectively at three collinear point.

> When $P$ is the Orthorcenter the the line is the Simson line of $P'$

> **Question again:** The generalization of the Simson line above is known?

[![enter image description here][2]][2]

**See also:**
* [Simson line](https://en.wikipedia.org/wiki/Simson_line)
* [Steiner line](http://users.math.uoc.gr/~pamfilos/eGallery/problems/SteinerLine.html)


  [1]: https://i.sstatic.net/iJgQh.png
  [2]: https://i.sstatic.net/WF9Yt.png