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Question
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.
Solution
Given in the question, D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equivalent ∆ABC.
To proof that ∆DEF is an equilateral triangle.
Proof: In ∆ABC, E and F are the mid-points of AC and AB respectively, then EF || BC.
So, EF = `1/2` BC ...(I)
DF || AC, DE || AB
DE = `1/2` AB and FD = `1/2` AC [By mid-point theorem] ...(II)
Now, ∆ABC is an equilateral triangle
AB = BC = CA
`1/2` AB = `1/2` BC = `1/2` CA ...[Dividing by 2 in the above equation]
So, DE = EF = FD ...[From equation (I) and (II)]
Since, all sides of ADEF are equal.
Hence, ∆DEF is an equilateral triangle
Hence proved.
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