In an equilateral triangle, centroid and the circumcentre coincide.
Consider an equilateral triangle △ABC inscribed in a circle.
Let O be the circumcentre.
Then OA = OB = OC - - -(1)
Draw three medians AD, BE and CF intersecting at G. Then G is the centroid of △ABC.
In △BFC and △CEB,
BC = BC; ∠B = ∠C = 60∘; BF = CE (since AB = AC)
∴ △BFC ≅ △CEB
∴ BE = CF, similarly, AD = BE
Thus AD = BE = CF
⟹ GA = GB = GC - - -(2) (∵ the centroid divides each median in the ratio 2:1)
Hence, from (1) and (2), we can conclude that G coincides with O.