In Figure, `D ,E` and `F` are, respectively the mid-points of sides `B C ,C A` and `A B` of an equilateral triangle `A B C` . Prove that `D E F` is also an equilateral triangle.
Let ABC be the triangle and D, E and F be the mid-point of BC, CA and AB respectively.
We have to show triangle formed DEF is an equilateral triangle. We know the line segment joining the mid-points of two sides of a triangle is half of the third side
Therefore,`DE=1/2AB`
`EF=1/2​BC`
`FD=1/2AC`
Now, `/_\ABC` is an equilateral triangle
`=>AB=BC=CA`
`=>1/2AB=1/2BC=1/2CA`
`=>DE=EF=FD`
`/_\DEF` is an equilateral triangle.

Explanation

Let ABC be the triangle and D, E and F be the mid-point of BC, CA and AB respectively.
We have to show triangle formed DEF is an equilateral triangle. We know the line segment joining the mid-points of two sides of a triangle is half of the third side
Therefore,`DE=1/2AB`
`EF=1/2​BC`
`FD=1/2AC`
Now, `/_\ABC` is an equilateral triangle
`=>AB=BC=CA`
`=>1/2AB=1/2BC=1/2CA`
`=>DE=EF=FD`
`/_\DEF` is an equilateral triangle.