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.
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<img src="http://cs.smith.edu/~orourke/MathOverflow/FivePointsr34.jpg" />
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Concerning the question, "Can the circle be generalized to any conic?":
certainly not straightforwardly.
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<img src="https://i.sstatic.net/K2ZCB.jpg" width="500" />
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***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:
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<img src="https://i.sstatic.net/HHm5S.jpg" width="500" />
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