Prove that the four triangles formed by joining in pairs, the mid-points of three sides of a triangle, are congruent to each other.

Explanation

A triangle of `ABC \and\ DEF` are the mid-points of sides BC,CA and AB respectively.
`/_\AFE~=/_\FBD~=/_\EDC~=/_\DEF.`
Since the segment joining the mid-points of the sides of a triangle is half of the third side.
`DE=1/2AB`
`=>DE=AF=BF`(1)
`EF= 1/2BC`
`=>EF=BD=CD` (2
`DF=1/2AC`
`=>DF=AE=EC`(3)
Now, in `/_\s DEF \and\ AFE`
`DE=AF`
`DF=AE`
`EF=FE`
So, by SSS criterion of congruence, `/_\DEF~=/_\AFE`
Similarly,`/_\DEF~=/_\FBD`
`/_\DEF~=/_\EDC`
Hence,`/_\AFE~=/_\FBD~=/_\ED~=/_\DEF`