hard geometry problem, hurry! one medal will be given for a solution+good explanation :)
$\large a,b,c,d$ are sides of a quadrilateral with all angles

Question

hard geometry problem, hurry! one medal will be given for a solution+good explanation :)
$\large a,b,c,d$ are sides of a quadrilateral with all angles < 180, and the length of no side/diagonal is greater than 1.

show that the preimeter is $ \le 2+4\sin(\pi/12)$

ganeshie8 · Answer

|dw:1407607357205:dw|

anonymous · Answer

go ahead @ganeshie8

ganeshie8 · Answer

I don't have a solution for this, i am still trying...

kainui · Answer

So the perimeter is $$\LARGE P \le 2+4\sin( \frac{\pi}{12})$$ is that right?

ganeshie8 · Answer

yes thats the thing we need to prove

kainui · Answer

ok, just making sure the latex or whatever wasn't really showing up.

ganeshie8 · Answer

prove   $$\large     a+b+c+d \le 2+4\sin( \frac{\pi}{12})$$

ganeshie8 · Answer

we're given that they are sides of a quadrilateral, and the lengths of sides and diagonals are <= 1

anonymous · Answer

are diagonals necessarily perpendicular to each other?

ganeshie8 · Answer

need not be, its just some random convex quadrilateral

kainui · Answer

Nope @k142 |dw:1407608424251:dw|

kainui · Answer

maybe I should have drawn |dw:1407608469320:dw| instead lol

anonymous · Answer

Haha! I got!

anonymous · Answer

but there must be a clue to start..

ganeshie8 · Answer

unfortunaltely no hints given :( i thought i figured out the solution while posting this problem, but i found a huge blunder in my work later!

kainui · Answer

Does having diagonals of length 1 necessarily imply that the other sides have to be less than or equal to 1 already?

kainui · Answer

Ahhh... no it doesn't. |dw:1407608821504:dw|

anonymous · Answer

no

anonymous · Answer

what was that blunder in your work?

ganeshie8 · Answer

|dw:1407609109876:dw|