Question

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

Answer 1

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 $DE=\frac{1}{2}AB,EF=\frac{1}{2}BC$ and $FD=\frac{1}{2}AC$

Now, $\Delta ABC$ is an equilateral triangle

$\Rightarrow AB=BC=CA$

$\Rightarrow \frac{1}{2}AB=\frac{1}{2}BC=\frac{1}{2}CA$

$\Rightarrow DE=EF=FD$

$\therefore \Delta DEF$ is an equilateral triangle.