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Prove that the medians of an equilateral triangle are equal.

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Consider an equilateral △ABC, and Let D, E, F are midpoints of BC, CA and AB.

Here, AD, BE and CF are medians of △ABC. 

Now, 

D is midpoint of BC => BD = DC 

Similarly, CE = EA and AF = FB 

Since ΔABC is an equilateral triangle 

AB = BC = CA …..(i) 

BD = DC = CE = EA = AF = FB …………(ii) 

And also, ∠ ABC = ∠ BCA = ∠ CAB = 60° ……….(iii) 

Consider Δ ABD and Δ BCE 

AB = BC [From (i)] 

BD = CE [From (ii)] 

∠ ABD = ∠ BCE [From (iii)] 

By SAS congruence criterion, 

Δ ABD ≃ Δ BCE 

=> AD = BE ……..(iv) [Corresponding parts of congruent triangles are equal in measure] 

Now, consider Δ BCE and Δ CAF, 

BC = CA [From (i)] 

∠ BCE = ∠ CAF [From (ii)] 

CE = AF [From (ii)] 

By SAS congruence criterion, 

Δ BCE ≃ Δ CAF 

BE = CF …………..(v) [Corresponding parts of congruent triangles are equal] 

From (iv) and (v), we have

AD = BE = CF 

Median AD = Median BE = Median CF 

The medians of an equilateral triangle are equal. 

Hence proved.

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