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April 22nd, 2018, 07:25 AM   #1
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Minimal value of the sum of two line segments

Quote:
 $\displaystyle ABCD$ is square with side $\displaystyle a$. There is a point E on side $\displaystyle AB$, so that $\displaystyle AE : EB = 2 : 1$. Let F be an arbitrary point on diagonal $\displaystyle BD$. Prove that $\displaystyle AF + EF \geqslant \frac{a \sqrt{10}}{3}$.
I proved that $\displaystyle AF + EF \geqslant \frac{a \sqrt{12}}{3}$.
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?
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Last edited by lua; April 22nd, 2018 at 07:27 AM. April 24th, 2018, 03:13 AM   #2
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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.
Attached Images picture.jpg (8.2 KB, 0 views)

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: 55 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. Tags line, minimal, segments, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post nomad609 Calculus 0 October 23rd, 2016 11:39 AM Jmlee19 Calculus 5 December 6th, 2012 02:23 PM master555 Applied Math 0 December 2nd, 2011 05:56 PM andoxx Algebra 7 February 8th, 2011 08:09 PM Cheesy74 Algebra 4 November 17th, 2009 05:23 AM

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