Question

points d e and f are the midpoints of the sides of triangle abc .point g is the centroid of triangle abc show that point g is the centroid os triangle def also

Answer 1

I am proving by coordinate geometry .

If you want geometric proof do comment !

In Δ ABC ,

Let A be x₁ , y₁

Let B be x₂ , y₂

Let C be x₃ , y₃

Mid point formula says :

D = x = ( x₁ + x₂ ) / 2  , y = ( y₁ + y₂ ) / 2

E = x = ( x₂ + x₃ ) / 2 , y = ( y₂ + y₃ ) / 2

F = x  = ( x₁ + x₃ ) / 2 , y = ( y₁ + y₃ ) / 2

Centroid of Δ ABC

x = ( x₁ + x₂ + x₃ ) / 3

y = ( y₁ + y₂ + y₃ ) / 3

Centroid of Δ DEF

x = [ ( x₁ + x₂ ) / 2 + ( x₁ + x₃ ) / 2 + ( x₂ + x₃ ) / 2 ] / 3

= > x = [ x₁ / 2 + x₂ / 2 + x₁ / 2 + x₃ / 2 + x₂ / 2 + x₃ / 2 ] / 3

Use  a / 2 + a / 2 = a

= > x = [ x₁ + x₂ + x₃ ] / 3

Similarly :

y = [ ( y₁ + y₂ ) / 2 + ( y₁ + y₃ ) / 2 + ( y₂ + y₃ ) / 2 ] / 3

= > y = [ y₁ / 2 + y₂ / 2 + y₁ / 2 + y₃ / 2 + y₂ / 2 + y₃ / 2 ] / 3

Use  a / 2 + a / 2 = a

= > y = [ y₁ + y₂ + y₃ ] / 3

Hence the coordinates of the centroid of Δ ABC is the same as the centroid of Δ DEF.

Therefore it is the same point !

[ P.R.O.V.E.D ]

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Answer 2

In Δ ABC ,

Let A be x₁ , y₁

Let B be x₂ , y₂

Let C be x₃ , y₃

Mid point formula says :

D = x = ( x₁ + x₂ ) / 2  , y = ( y₁ + y₂ ) / 2

E = x = ( x₂ + x₃ ) / 2 , y = ( y₂ + y₃ ) / 2

F = x  = ( x₁ + x₃ ) / 2 , y = ( y₁ + y₃ ) / 2

Centroid of Δ ABC

x = ( x₁ + x₂ + x₃ ) / 3

y = ( y₁ + y₂ + y₃ ) / 3

Centroid of Δ DEF

x = [ ( x₁ + x₂ ) / 2 + ( x₁ + x₃ ) / 2 + ( x₂ + x₃ ) / 2 ] / 3

= > x = [ x₁ / 2 + x₂ / 2 + x₁ / 2 + x₃ / 2 + x₂ / 2 + x₃ / 2 ] / 3

Use  a / 2 + a / 2 = a

= > x = [ x₁ + x₂ + x₃ ] / 3

Similarly :

y = [ ( y₁ + y₂ ) / 2 + ( y₁ + y₃ ) / 2 + ( y₂ + y₃ ) / 2 ] / 3

= > y = [ y₁ / 2 + y₂ / 2 + y₁ / 2 + y₃ / 2 + y₂ / 2 + y₃ / 2 ] / 3

Use  a / 2 + a / 2 = a

= > y = [ y₁ + y₂ + y₃ ] / 3

Hence the coordinates of the centroid of Δ ABC is the same as the centroid of Δ DEF.