Prove that medians of equilateral triangle are equal - Maths - Triangles

Question

Prove that medians of equilateral triangle are equal

Harleen Birdi · Accepted Answer

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 $\Rightarrow {AD}^{2} = {AB}^{2} - {BD}^{2} = x^{2} - (\frac{x}{2})^{2} = \frac{3 x^{2}}{4} [AD is the median BD = \frac{1}{2} BC = \frac{x}{2}] \Rightarrow AD = \frac{\sqrt{3} x}{2}$ In right ΔBEC, BC2 = BE2 + CE2

Similarly, we can show that CF = 3x2 Thus, AD = BE = CF Hence we are done

Prove that medians of equilateral triangle are equal.