lf D, E and F are respectively the midpoints of the sides BC, CA and AB of an equilateral triangle ABC, prove that `triangle DEF` is also an equilateral triangle.
Text Solution
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Since D and F are the midpoints of sides BC and CA respectively, we have `DE=(1)/(2)AB` (by midpoint theorem). Similarly, `FE=(1)/(2) BC and DF =(1)/(2) BC and DF =(1)/(2) CA. ` `therefore AB =BC = CA rArr (1)/(2) AB =(1)/(2) BC =(1)/(2)CA rArr DE = FE = DF.` Thus, all the sides of `triangle DEF` are equal . Hence, ` triangle DEF ` is an equilateral triangle.
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