Question

In the adjoining figure D, E and F are the mid-points of the sides BC, CA and AB of the equilateral `DeltaABC.` Prove that `DeltaDEF` is also an equilateral triangle.

Answer 1

In `triangleABC,`
F is the mid-point of AB
E is the mid point of AC
`therefore EF=(1)/(2)BC`(mid point theorem)…(1)
Similarly,
`FD=(1)/(2)AB" "`(mid point theorem)`" "`…(2)
and `ED=(1)/(2)AB" "`(mid-point theorem)`" "`...(3)
but AB =BC =CA
`implies (1)/(2)AB =(1)/(2)BC=(1)/(2)CA`
`implies ED=EF=FD`
`implies triangleDEF` is an equilateral triangle. Hence proved.