Find the smallest expression value

Oct 2019
6
0
Ukraine
About a convex hexagon ABCDEF it is known that AB = BC, CD = DE, EF = FA. Find the smallest value of the expression: BC / BE + DE / DA + FA / FC
 
Oct 2019
6
0
Ukraine
yes, but how to prove that this value will be the smallest?
 
Mar 2015
182
68
Universe 2.71828i3.14159
By proving it gets smallest number if all sides are equal(?)
 
Oct 2019
6
0
Ukraine
This is understandable, but how to prove that the correct hexagon will have the minimum value, and will any other have a larger value?
 
Mar 2015
182
68
Universe 2.71828i3.14159
The sides are given, so we need maximum values for diagonals.
Draw diagonals $AC, CE, EA$ and we get 3 isosceles triangles. H is a point on AC such that, BH and AC are perpendicular. The maximum value BE can get is EH+BH (for a triangle sum of two sides is greater than 3rd) and it's only possible if ABCE is a kite, so AE=EC.
Do it for other diagonals (DE, FA) and you will get the answer.