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.

  • The way to start seems obvious: express the sides of your triangle using vectors, then work to express the medians as vectors, then see what the hypothesis gives you. – rschwieb Jan 7 '16 at 16:33
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    Apollonius' theorem in effect gives $m_a^2=\frac14(2b^2+2c^2-a^2)$ and $m_b^2=\frac14(2a^2+2c^2-b^2)$ which makes it very easy to show $m_a=m_b \implies a=b$. Apollonius' theorem can be proved with vectors and the diagonals of a parallelogram – Henry Jan 7 '16 at 16:40
  • What do you know about vectors? It will not be much use to answer the question if the answer relies on facts you don't know yet. – David K Jan 7 '16 at 18:50

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.

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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.

  • The vectors of the two medians are $-2v+w$ and $-2w+v$. – user236182 Jan 7 '16 at 18:12
  • I have edited it. – user236182 Jan 7 '16 at 18:14
  • @user236182 Thank you! – mrprottolo Jan 7 '16 at 18:20

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