Prove that if the lengths of two medians of a triangle are equal, then the triangle is isosceles.

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Solution

Let the medians $A D$ and $B E$ of the triangle $△ A B C$ be equal.

We know that the lengths of medians are given by $A D^{2} = \frac{1}{4} (2 b^{2} + 2 c^{2} - a^{2})$ and $B E^{2} = \frac{1}{4} (2 c^{2} + 2 a^{2} - b^{2})$

$A D = B E ⟹ 2 b^{2} + 2 c^{2} - a^{2} = 2 c^{2} + 2 a^{2} - b^{2}$
$⟹ b^{2} = a^{2} ⟹ b = a$

Thus $△ A B C$ is an isosceles triangle

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