PROVE that medians of an 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,

AB^{2} = AD^{2} + BD^{2}

In right ΔBEC,

BC^{2} = BE^{2} + CE^{2}

Similarly, we can show that CF = .

Thus, AD = BE = CF.

hence proved