Polygons, Centroids & Circumcentres!

$X$ is a polygon with $n [\geq3]$ sides. The centroid and the circumcentre of $X$ happens to be the same point. Read the following statements about $X$

$[1]$ . If $n=3$ , then $X$ is a regular polygon.

$[2]$ If $n=4$ , then $X$ is not necessarily a regular polygon.

$[3]$ . $X$ is a regular polygon whenever $n\neq 4$ .

Which of these statements are correct?

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This problem is from the set "MCQ Is Not As Easy As 1-2-3". You can see the rest of the problems here .