MHB When Are the Equivalence and Inequality Met?

[Albert1]

Rectangle ABCD ,having fours points$ E,F,G,H $ located on segments AB, BC, CD,and

DA respectively , please prove :

$EF+FG+GH+HE\geq 2 AC $

and determine when the equivalence can be taken ?

 

[Albert1]

Re: Ef+fg+gh+he>=2ac

suppose we have another three cards equivalent to figure 1 ,and rearranging these four cards
in a position as shown in figure 2
from figure 2 we see :2AC=AP=HQ< HG+GM+MN+NQ=EF+FG+GH+HE
figure 3 shows the equivalence will be taken when E,F,G,H are midpoints of AB,BC,CD,and DA respectively
2AC=AC+BD=EF+FG+GH+HE
View attachment 1644