Give △ABC is acute triangle. M is a midpoint of BC Prove that AB+AC>2AM


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Question Number 199481 by tri26112004 last updated on 04/Nov/23

$G i v e △ A B C i s a c u t e t r i a n g l e .$ $M i s a m i d p o i n t o f B C$ $P r o v e t h a t A B + A C > 2 A M$
	Answered by mr W last updated on 04/Nov/23

	Commented by mr W last updated on 04/Nov/23

	$A B + {B A}^{'} > {A A}^{'}$ $\Rightarrow A B + A C > 2 A M$
	Commented by tri26112004 last updated on 04/Nov/23

	$y o u h a v e o t h e r s o l u t i o n ¿$
	Commented by mr W last updated on 04/Nov/23

	$t h i s i s t h e s i m p l e s t s o l u t i o n .$
	Commented by mr W last updated on 04/Nov/23

	$a n o t h e r s o l u t i o n :$ $B C + A C > A B \Rightarrow B C > A B - A C$ $2 A M = \sqrt{2 {A B}^{2} + 2 {A C}^{2} - {B C}^{2}}$ $< \sqrt{2 {A B}^{2} + 2 {A C}^{2} - {(A B - A C)}^{2}}$ $= \sqrt{{(A B + A C)}^{2}}$ $= A B + A C$
	Commented by tri26112004 last updated on 05/Nov/23

	$G o o d s o l u t i o n$
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