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 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 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