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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

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

See also:

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    $\begingroup$ This is one those problems that can be solved using the so-called $p,q$ method, after two reductions: 1) by affine invariance, we cann suppose that the circumconic is, in fact, the circumcentre. 2) we can assume that the vertices are $(0,0)$, $(1,0)$ and $(p,q)$ . We can then compute the coordinates of the auxiliary points and so apply the standard collinearity condition that the area of the corresponding triangle be zero. $\endgroup$
    – hordubal
    Commented Oct 10, 2021 at 11:59
  • $\begingroup$ Yes, the solution maybe not hard. But my question that is it known? $\endgroup$
    – Đào Thanh Oai
    Commented Oct 10, 2021 at 12:02
  • $\begingroup$ Sorry— misunderstood. No idea if it is known. $\endgroup$
    – hordubal
    Commented Oct 10, 2021 at 12:03
  • $\begingroup$ Thank you for your comment. $\endgroup$
    – Đào Thanh Oai
    Commented Oct 10, 2021 at 12:04
  • $\begingroup$ The simson line is very well-known, and have some nice properties, I hope that the line will know and have some nice properties $\endgroup$
    – Đào Thanh Oai
    Commented Oct 10, 2021 at 12:05

1 Answer 1

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This is not an answer, but I'd like to point out that the concepts in question are projective, although they have a special Euclidean case.

Consider the diagram below. Start with a conic $\gamma$(green), a triangle $ABC$ inscribed in $\gamma$, and a line $\omega$ (black dot-dashed). Let $X$ be the polar of $\omega$ wrt the conic, and draw a line (dotted) through $X$ that meets $\omega$ at $P,P'$. Let the dashed lines through $P'$ meet the respective lines from $P$ to $A,B,C$ at $\omega$. Then the dashed lines meet the triangle sides at collinear points (red), and $X$ lies on this line.

The OP is the special case when $\omega$ is the projective line at infinity.

So, if the line in OP Question 1 is known, and anybody is trying to hunt it down, it may be in the projective geometry literature.

enter image description here

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