Given a triangle ABC and D is the midpoint of BC. Prove that

(a) AD < (AB+AC)/2

(b) AD > (AB+AC-BC)/2

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- Sep 5th 2011, 07:10 AMMichaelLightBasic geometry proof
Given a triangle ABC and D is the midpoint of BC. Prove that

(a) AD < (AB+AC)/2

(b) AD > (AB+AC-BC)/2 - Sep 5th 2011, 07:35 AMPlatoRe: Basic geometry proof
Using vectors this is rather easy.

$\displaystyle \overrightarrow {DA} = \overrightarrow {AB} + \overrightarrow {BD} \;\& \,\overrightarrow {DA} = \overrightarrow {AC} + \overrightarrow {CD} $

Note that $\displaystyle \overrightarrow {BD}+\overrightarrow {CD}=0$.

Use the triangle inequality.