The midpoint of one diagonal in a parallelogram is also the midpoint of another [closed]

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If ABCD is a parallelogram and M is the midpoint of AC, prove that M is also the midpoint of BD.

I think vectors must be used to solve this problem

I know that this is part of the definition of a parallelogram but this question is assuming that you don't know that and therefore proving it's true.

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2 Answers

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Hint (from the figure)

$$ DM=MB \qquad and \qquad CM=AM $$

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See emilios figure so i continue from there sorry dunno how to add pictures.$|DM|+|MC|=|DC|$.(1).(triangle law of vectors. And $|AM|+|MB|=|AB|$..(2) but $|AB|=|DC|$. .. opposite sides of parallelogram are equal thus $|DM|+|MC|=|AM|+|MB|$.. from 1,2 but$|AM|=|MC|$ M is the midppint thus $|DM|=|MB|$ which implies M is also the midpoint of diagonal BD.

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