Question

In an equilateral triangle prove that the centroid and the Circumcircle of the triangle coincide.

•Hint:-
Prove that the medians are perpendicular bisectors of the sides of triangle.
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Answer 1

$\sf \large \underbrace{ \underline{Understanding \: the \: Question}}$

In this question we have to prove that centroid and circumcircle of triangle coincidence which means that centroid G is equidistant from vertices. Which we can simplify prove by using congruency.

First we should know the property of equilateral ∆

• All sides are equal in equilateral∆
• Also all the angles are of 60°

Also we know that median of a triangle divides the side in equal length.

So:-

• BD=DC
• BF=AF
• AE=EC

So let's apply congruency rule in ∆BCF and ∆BCE

BF=CE (Proved above)

angle B=Angle C ( each 60°)

BC=BC (Common)

So ∆BCF is congruent to ∆BCE (by SAS)

BE=CF( by C.P.C.T).....(1)

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Now apply congruency rule in ∆AFC and ∆ADC

AF=DC (Proved above)

Angle A =Angle C ( each 60°)

AC=AC (Common)

So ∆AFC is congruent to ∆ADC (By SAS)

AD=CF (by C.P.C.T)......(2)

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From (1) and (2) equations

BE=CF=AD.....(3)

As we know that centroid of the triangle (G) divide the median in the ratio 2:3.

So on dividing the equation (3) by 2/3 we get:-

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

GB=GC=GA

Now we can say that centroid G is equidistant from the three vertices of ∆ ABC.

Hence it is proved that G is centroid as well as centre of circumcircle of equilateral triangle or Centroid and circle are coincide.

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$\Large \sf \underbrace{ \underline{More \: To \: Know!}}$

★What is centroid?

➠The point at which three medians of the triangle meet is called it's centroid.

★What is circumcircle?

➠It is the special type of circle in which triangle is inscribed.

★What is C.P.C.T.?

➠Corresponding part of congruent triangle.

Answer 2

Answer:

Given: An equilateral triangle ABC in which D, E and F are the mid-point of sides BC, CA and AB respectively. To prove: the centroid and circumcentre are coincident. Read more on Sarthaks.com - https://www.sarthaks.com/268596/equilateral-triangle-prove-that-the-centroid-centre-circum-circle-circumcentre-coincide