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?

**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?

**See also:**

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