Question

In a equilateral triangle , prove that the centroid and the CircumCircle (Circumcentre) respectively.​

Answer 1

Question

In a equilateral triangle , prove that the centroid and the CircumCircle (Circumcentre) respectively.​

Answer:

Given : An equilateral triangle ABC in which D, E and F are the mid- points of sides BC,

CA and AB respectively.

To prove : The centroid and circumcentre  are coincident.

Construction : Draw medians, AD, BE and CF.

Proof : Let G be the centroid of  ΔABC  i. e., the point of intersection of AD, BE and CF. In triangles BEC and BFC, we have

∠B = ∠C = 60

BC = CE     [∵  AB = AC  ⇒ ½ AB = ½ AC ⇒ BF = CE]

∴   ΔBEC ≅ ΔBFC

⇒ BE = CF     ……(i)

Similarly,   ΔCAF ≅  ΔCAD

CF = AD        …….(ii)

From (i) & (ii)   AD = BE = CF

⇒  2/3 AD = 2/3, BE = 2/3 CF

CG = 2/3 CF

GA = 2/3 AD, GB = 2/3 BE

⇒ GA = GB = GC

⇒ G is equidistant from the vertices

⇒ G is the circumcentre of  ΔABC

Hence, the centroid and circumcentre are coincident.

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