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As this construction was new to me, I wanted to see it. Here are two random examples. The green points are centroids of the triples of points.

Concerning the question, "Can the circle be generalized to any conic?": certainly not straightforwardly.

Update. Here is an illustration of Aaron Meyerowitz's beautiful theorem, for $N=8$, $m=2$, showing coincidence of the $\binom{8}{2}=28$ lines through the $n=6$ hexagon centroids $q$ (green) and (in this case) perpendicular to the line through the two complementary points:

As this construction was new to me, I wanted to see it. Here are two random examples. The green points are centroids of the triples of points.

Concerning the question, "Can the circle be generalized to any conic?": certainly not straightforwardly.

Update. Here is an illustration of Aaron Meyerowitz's beautiful theorem, for $N=8$, $m=2$, showing coincidence of the $\binom{8}{2}=28$ lines through the $n=6$ hexagon centroids $q$ (green) and (in this case) perpendicular to the line through the two complementary points:

As this construction was new to me, I wanted to see it. Here are two random examples. The green points are centroids of the triples of points.

Concerning the question, "Can the circle be generalized to any conic?": certainly not straightforwardly.

Update. Here is an illustration of Aaron Meyerowitz's theorem, for $N=8$, $m=2$, showing coincidence of the $\binom{8}{2}=28$ lines through the hexagon centroids $q$ (green) and (in this case) perpendicular to the line through the two complementary points:

As this construction was new to me, I wanted to see it. Here are two random examples. The green points are centroids of the triples of points.

Concerning the question, "Can the circle be generalized to any conic?": certainly not straightforwardly.

Update. Here is an illustration of Aaron Meyerowitz's beautiful theorem, for $N=8$, $m=2$, showing coincidence of the $\binom{8}{2}=28$ lines through the $n=6$ hexagon centroids $q$ (green) and (in this case) perpendicular to the line through the two complementary points:

As this construction was new to me, I wanted to see it. Here are two random examples. The green points are centroids of the triples of points.

Concerning the question, "Can the circle be generalized to any conic?": certainly not straightforwardly.

As this construction was new to me, I wanted to see it. Here are two random examples. The green points are centroids of the triples of points.

Concerning the question, "Can the circle be generalized to any conic?": certainly not straightforwardly.

Update. Here is an illustration of Aaron Meyerowitz's theorem, for $N=8$, $m=2$, showing coincidence of the $\binom{8}{2}=28$ lines through the hexagon centroids $q$ (green) and (in this case) perpendicular to the line through the two complementary points: