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

Given,

△ABC is an equilateral triangle.

⇒ AB = BC = AC

By mid-point theorem,

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Since, D and E are the mid-points of AB and BC respectively.

⇒ DE = $\dfrac{1}{2}$ × AC

⇒ DE = $\dfrac{1}{2}$ × AB [As AB = AC = BC] ….(1)

Since, D and F are the mid-points of AB and AC respectively.

⇒ DF = $\dfrac{1}{2}$ × BC

⇒ DF = $\dfrac{1}{2}$ × AB [As AB = AC = BC] ….(2)

Since, E and F are the mid-points of BC and AC respectively.

⇒ EF = $\dfrac{1}{2}$ × AB ….(3)

From eq.(1), (2) and (3), we have:

⇒ DE = DF = EF

∴ △DEF is an equilateral triangle.

Hence, proved that △DEF is an equilateral triangle.