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

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

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Solution.

D and E are midpoints of BC and AC respectively, so using mid-point theorem:
\Rightarrow DE=\frac{1}{2}AB                             …..(1)
E and F are midpoints of AC and AB, so using mid-point theorem:
\Rightarrow EF=\frac{1}{2}BC                           …..(2)
F and D are midpoints of AB and BC, so using mid-point theorem:
\Rightarrow FD=\frac{1}{2}AC                          …..(3)
It is given that \triangle ABC is an equilateral triangle
\Rightarrow AB=BC=CA
\frac{1}{2}AB=\frac{1}{2}BC=\frac{1}{2}CA           {dividing by 2}
Using 1, 2 and 3 we get
DE = EF = FD
Hence DEF is an equilateral triangle.
Hence proved

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