Question

The centroid of an equilateral triangle coincides with which of the following options?

• A. Circumcentre
• B. Orthocentre
• C. Incentre
• D. Altitude

Answer 1

Answer: (A), (B), (C)

The correct options are
A Circumcentre
B Orthocentre
C Incentre

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; $\angle$B = $\angle$C = 60${}^{\circ }$; BF = CE (since AB = AC)

$\therefore$ $△$BFC $\cong$ $△$CEB

$\therefore$ BE = CF, similarly, AD = BE

Thus AD = BE = CF

$⟹\frac{2}{3}AD=\frac{2}{3}BE=\frac{2}{3}CF$

$⟹$ GA = GB = GC - - -(2) ($\because$ the centroid divides each median in the ratio 2:1)

Hence, from (1) and (2), we can conclude that G coincides with O.
Similarly, the centroid of an equilateral triangle coincides with orthocentre and incentre also.