Question

# In an equilateral triangle, centroid and the circumcentre coincide.

A

True

B

False

Solution

## The correct option is A True 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 ⟹23AD=23BE=23CF ⟹ 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.

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