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?

Remarks: Let circumconic is circumcircle and $X=$ Steiner line respect to $P$ $\cap$ the new line respect to $P$, then locus of $X$ is an ellipse through the Orthocenter and Circumcenter.

See also:

• Simson line
• Steiner line