Question

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

Answer 1

Triangle ABC is an equilateral triangle.
⇒ AB = BC = AC
DE = 1/2 AB

EF = 1/2 BC
⇒ EF = 1/2 AB [Since AB = BC = AC]

DF = 1/2 AC
⇒ DF = 1/2 AB [Since AB = BC = AC]

DE = EF = DF

∴ ΔDEF is an equilateral triangle.

Answer 2

Given:

D, E, and F are the midpoints of the sides BC, CA, and AB respectively of an equilateral ΔABC.

To Find:

ΔDEF is an equilateral triangle.

Solution:

According to the question,  AF = BF, BD = CD, AE = CE

As we know, A = BC = CA

Now, in ΔABC,

F and D are midpoints using the middle point theorem.

FD ║ AC

FD = 1/2 AC ..(i)

Then, in ΔABC, F and E are the midpoints

FE ║ BC

FE = 1/2BC ..(ii)

Now, in ΔABC, E and D are the midpoints.

AB ║ ED

DE = 1/2AB ..(iii)

Using (i), (ii) and (iii)

FD = FE = DE  = 1/2AC = 1/2BC = 1/2AB

FD = FE = DE = 1/2AC  [∵AB = BC = AC]

So, ΔDEF is an equilateral triangle.