Is the circumcenter of an equilateral triangle equidistant from its 3 vertices? If yes, how can I prove that?
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Yes, that's true for any triangle.
Since the centre of a cirle is equidistant from all point on the circle, it follows that the circumcentre is equidistant from the vertices.
In the case of an equilateral triangle, all the geometrical centres of the triange coincide. That means that not only the circumcentre, but the incentre, orthocentre and centroid are equidistant from the vertices too.
This isn't the case for other triangles.
Yes. The circumcenter is the center of the circle passing through the vertices. So it is exactly at a distance $r$ away from the vertices. $r$ is the radius of the circumcircle.