Q. 23.5( 10 Votes )
If the length of three medians of a triangle are equal. Let us prove that the triangle is an isosceles triangle.
Answer :
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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