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If two medians of a triangle are equal then prove by vector method that it is an isosceles $triangle$ This might be a simple question but i could not do it because i don't know any theorems related to vector.
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$\begingroup$I'm not sure if this is what you're looking for. Name $v,w$ the two vectors in the directions of the two sides of the triangle with half of the lenght.
The vectors of the two medians can be expressed by $-2v+w$ and $-2w+v$. They are equal in lenght, so $$|-2v+w|=|-2w+v|$$ that can be rewritten using dot product \begin{align} (-2v+w)\cdot (-2v+w) &=(-2w+v)\cdot (-2w+v) \\ 4|v|^2-4w\cdot v+|w|^2 &= 4|w|^2-4w\cdot v+|v|^2 \\ 3|v|^2&=3|w|^2. \end{align} Hence $|v|=|w|$ and the triangle is isosceles.
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