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If we have a triangle like so where $\overline{AM}$ is a median, can we say something about the ratios of $|\overline{AB}|$, $|\overline{AM}|$, and/or $|\overline{AC}|$? Specifically, is it true by any chance that $$\frac{|\overline{AB}| + |\overline{AC}|}{2} \ge |\overline{AM}|$$ holds. (I realize this is probably basic geometry, but I haven't done said geometry for a long time and it was never my strong suit.)

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kanso37
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    Yes, that is true. Denoting $a=|BC|,b=|CA|,c=|AB|$ and $m=|AM|$ the result $2m=\sqrt{2b^2+2c^2-a^2}$ can easily be used to prove it. – almagest Jan 29 '20 at 19:06

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Using vectors, we have that

$$ AM = AB + BM \hskip3ex \text{and} \hskip3ex AM = AC + CM $$

from which, adding up, we get

$$ 2AM = AB + AC +(BM - MC) = AB + AC $$

because $BM = MC.$ Taking the norms, finally we get

$$ 2|AM| = |AB + AC| \leq |AB| + |AC|. $$

gpassante
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