Question

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

Answer 1

Given in the question, D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equivalent ∆ABC.

To proof that ∆DEF is an equilateral triangle.

Proof: In ∆ABC, E and F are the mid-points of AC and AB respectively, then EF || BC.

So, EF = `1/2` BC  ...(I)

DF || AC, DE || AB  

DE = `1/2` AB and FD = `1/2` AC   [By mid-point theorem]  ...(II)

Now, ∆ABC is an equilateral triangle

AB = BC = CA

`1/2` AB = `1/2` BC = `1/2` CA  ...[Dividing by 2 in the above equation]

So, DE = EF = FD   ...[From equation (I) and (II)]

Since, all sides of ADEF are equal.

Hence, ∆DEF is an equilateral triangle

Hence proved.