In an equilateral triangle, prove that the centroid and centre of the circum-circle (circum centre) coincide) Given : An equilateral triangle `A B C` in which `D ,\ E\ a n d\ F` are the mid-points of sides `B C ,\ C A\ a n d\ A B` respectively. To Prove: The centroid and circum centre are coincident. Construction : Draw medians `A D ,\ B E\ a n d\ C F`

Explanation

let `G` is the centroid of `/_\ ABC.`
In `/_BEC and /_BFC`
`/_B=/_C=60^@`
`BC=BC,BF=CE`
`/_\BEC~=/_\BFC`
`BE=CF`.....(1)
Similarly ,`/_\CAF~=/_\CAD`
`CF=AD`.....(ii)
From (i) and (ii),
`AD=BE=CF`
`2/3AD=2/3BE=2/3CF`
`GA=GB=GC`
`G` is equidistant form the vertices.
`G` is the circumstances of of `/_\ ABC`