In an equilateral triangle, prove that the centroid and the centre of the circum circle (circumcentre) coincide.

An equilateral ΔABC such that AB = BC = AC
To Prove: The centroid and circum-centre of ΔABC coincide.
Construction: Draw three medians AD, BE and CF intersecting at G. Then G is the centroid of ΔABC.
Proof: In ΔBFC and ΔCEB,
BC = BC, ∠B = ∠C = 60°
BF = CE  (AB = AC)
.^..  D BFC @ D CEB
.^..  BE= CF, similarly, AD = BE
Thus, AD = BE = CF
⇒ CF
⇒ GA = GB = GC  (^..^. the centroid divides each median in the ratio 2 : 1)
Also OA = OB = OC, where O is the circum-centre.
Hence, G coincides with O.