Question

Prove that the triangle formed by joining the midpoints of an equilateral triangle is also an equilateral

Answer 1

Let DEF be the midpoints of sides of a triangle ABC( with D on BC, E on AB and F on AC ). Now, considering triangles AEF and ABC, angles EAF = BAC and AE / AB = 1/2 and AF/AC = 1/2. 

Hence, both triangles are similar by the SAS ( Side - Angle - Side ) criterion and correspondingly as AE/AB=AF/AC=EF/BC ( similar triangle properties ), EF =BC/2.

the cases DF=AC/2 and DE=AB/2 can be proved in the same way. 

So,

AB=BC=AC (from the given data)

2DF=2EF=2DE

DE=EF=DF

So triangle DEF is also Equilateral Triangel

The triangle formed by joining the mid-points of the equilateral triangle is also an equilateral triangle