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April 22nd, 2018, 07:25 AM  #1  
Member Joined: Jan 2018 From: Belgrade Posts: 50 Thanks: 2  Minimal value of the sum of two line segments Quote:
It's the case when $\displaystyle AF \geqslant \frac{AC}{2}$, and $\displaystyle EF \geqslant \frac{a}{3\sqrt{2}}$ (when $\displaystyle EF \perp BD$). Do you know how to prove the original inequality? Last edited by lua; April 22nd, 2018 at 07:27 AM.  
April 24th, 2018, 03:13 AM  #2 
Senior Member Joined: Feb 2010 Posts: 694 Thanks: 132 
Reflect point $\displaystyle E$ around diagonal $\displaystyle BD$ to hit side $\displaystyle BC$ at point $\displaystyle G$. Connect $\displaystyle AG$ hitting the diagonal at point $\displaystyle F$. Now $\displaystyle AF+EF=AF+FG=AG$. By the Pythagorean Theorem $\displaystyle AG = \dfrac{a}{3}\sqrt{10}$. Sorry my picture is upside down from yours. Last edited by mrtwhs; April 24th, 2018 at 03:21 AM. 
April 24th, 2018, 04:51 AM  #3 
Member Joined: Jan 2018 From: Belgrade Posts: 50 Thanks: 2 
O, great. I see. If $\displaystyle F$ was anywhere else on $\displaystyle BD$, it would be: $\displaystyle AF+EF=AF+FG>AG$. So, $\displaystyle AG$ is minimal value of $\displaystyle AF+EF$. Thank you, very much. 

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line, minimal, segments, sum 
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