Question

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

Answer 1

Since line segment joining the mid-points of two sides of a triangle is half of the third side. Therefore, D and E are mid-points of BC and AC respectively.

⇒ DE = 1 / 2 AB --- (i)

E and F are the mid - points of AC and AB respectively .

∴ EF = 1 / 2 BC --- (ii)

F and D are the mid - points of AB and BC respectively .

∴ FD = 1 / 2 AC --- (iii)

Now, △ABC is an equilateral triangle .

⇒ AB = BC = CA

⇒ 1 / 2 AB = 1 / 2 BC = 1 / 2 CA

⇒ DE = EF = FD [using (i) , (ii) , (iii) ]

Hence, DEF is an equilateral triangle .