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In an equilateral triangle prove that th...

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

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The point of intersection of the medians of a triangle is called its centroid. The centroid of a triangle is the point located at `(2)/(3)` of the distance from a vertex along a median. The center of the circumcircle of this triangle is called the circumcentre.
Let `Delta ABC` be the given equilateral triangle and let its medians `AD, BE` and `CF` intersect at `G`.
Then `G` is the centroid of `Delta ABC`.
In `Delta BCE` and `Delta CBF` we have
`BC` `=` `CB` [common]
`/_ B = /_ C `[ each equal to `60^(@) ]`
`CE = BE [ AC = AB implies (1)/(2) AC =(1)/(2)AB]`
`:. Delta ABC =CF`
Similarly, `AD=BE`.
Thus, `AD=BE=CFimplies(2)/(3)AC=(1)/(2)AB]`
`implies GA=GB=GC`
This shows that `G` is the circumcentre of `Delta ABC`
Hence, `G` is the centroid as well as circumcentre of `Delta ABC`.
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