In this chapter we will prove that median of equilateral triangle are equal in length.

Given below is an equilateral triangle in which AB = BC= CA

Lines AD, BE & CF are the medians of the triangle.

Median divide the side into equal halves.

AF = FB

BD = DC

AE = EC

**To prove:**

All the medians are of equal length.

AD = BE = CF

**Solution**

Since all sides are equal, the half length of given sides are also equal.

AF = FB = BD = DC = CE = EA

Now consider triangle ABD and BCE;

AB = BC

∠B = ∠C = 60 degree ( equilateral triangle)

BD = CE

By SAS congruency, both the triangles are congruent.

Hence AD = BE

Now consider triangle BCE and CAF

BC = CA

∠C = ∠A

CE = AF

By SAS congruency, both the triangles are congruent.

Hence, BE = CF

Combining all the results we get;

AD = BE = CF

Hence inn equilateral triangle, all the medians are of equal length.