# If the length of three medians of a triangle are equal. Let us prove that the triangle is an isosceles triangle.

In Δ BGD and Δ CGE,

BGD = CGE (vertically opposite angles) …(1)

BE = DC (medians are equal)

Since, centroid divides the median in ratio 2:1

So, and

BG = CG (as BE = DC) …(2)

And and

GE = DG (BE = DC) …..(3)

Hence, by SAS congruency, Δ BGD and Δ CGE are congruent.

By CPCT, BD = EC

2× BD = 2 × EC

AB = AC

Hence, the triangle is isosceles.

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