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In the adjoining figure, ∆ABC is an equilateral triangle. Points F, D and E are midpoints of side AB, side BC, side AC respectively. Show that ∆FED is an equilateral triangle.

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Given: ∆ABC is an equilateral triangle. 

Points F, D and E are midpoints of side AB, side BC, side AC respectively. 

To prove: ∆FED is an equilateral triangle. 

Proof: 

∆ABC is an equilateral triangle. [Given] 

∴ AB = BC = AC ….(i) [Sides of an equilateral triangle] 

Points F, D and E are midpoints of side AB and BC respectively.

∴ FD = (1/2)AC …..(ii) [Midpoint theorem] 

Points D and E are the midpoints of sides BC and AC respectively. 

∴ DE = (1/2)AB …..(iii) [Midpoint theorem] 

Points F and E are the midpoints of sides AB and AC respectively. 

∴ FE = (1/2) BC 

∴ FD = DE = FE [From (i), (ii), (iii) and (iv) ] 

∴ ∆FED is an equilateral triangle.

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