$D, E$ and $F$ are the mid-points of the sides $BC, CA$ and $AB$ respectively of an equilateral triangle $ABC$. Show that $\triangle DEF$ is also an equilateral triangle.

(N/A) Given: $\triangle ABC$ is an equilateral triangle. $D, E$ and $F$ are the mid-points of the sides $BC, CA$ and $AB$ respectively of $\triangle ABC$.
To prove: $\triangle DEF$ is an equilateral triangle.
Proof: By the Mid-point Theorem,the line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.
$1$. $EF$ joins the mid-points of sides $AB$ and $AC$. Therefore,$EF = \frac{1}{2} BC$ $(1)$.
$2$. $DE$ joins the mid-points of sides $BC$ and $AC$. Therefore,$DE = \frac{1}{2} AB$ $(2)$.
$3$. $DF$ joins the mid-points of sides $BC$ and $AB$. Therefore,$DF = \frac{1}{2} AC$ $(3)$.
Since $\triangle ABC$ is an equilateral triangle,we have $AB = BC = CA$ $(4)$.
Substituting $(4)$ into $(1), (2)$ and $(3)$,we get:
$EF = \frac{1}{2} BC$,$DE = \frac{1}{2} BC$,and $DF = \frac{1}{2} BC$.
Thus,$DE = EF = DF$.
Therefore,$\triangle DEF$ is an equilateral triangle. Hence proved.