Prove that medians of equilateral triangle are equal.
Consider an equilateral ΔABC. Let AD, BE and CF be the medians of BC, AC and AB respectively.
We know that in an equilateral Δ, medians are altitudes as well.
Let AB = BC = AC = x [say]
In right ΔABD, by pythagoras theorem,
AB2 = AD2 + BD2
In right ΔBEC,
BC2 = BE2 + CE2
Similarly, we can show that CF = .
Thus, AD = BE = CF.
Hence we are done.