In ΔABC, medians BD and CE are equal and intersect each other at O. Prove that ΔABC is an isosceles triangle.

We know that the medians of a triangle intersect each other in the ratio 2 : 1.

BD = CE (Given)

2/3 BD =2/3 CE

⇒ OB = OC

1/3 BD =1/3 CE

⇒ OE = OD

In ΔBOE and ΔCOD,

OB = OC

OE = OD

∠BOE = ∠COD (Vertically opposite angles)

ΔBOE = ΔCOD (By SAS rule)

BE = CD (Congruent parts of congruent triangles)

2BE = 2CD

⇒ AB = AC

Hence ΔABC is an isosceles triangle.