hey u all ppl i have got the shortest method!!!
one which shakir wanted!!
we take triangle ABC and we take angles in it .
angle A=pi/4 we take

B=2

C=2

we now construct the angle bisectors of all sides and let then intersect at O
side BC =5(cot

+cot

)=10sin(

+

)/( sin(

+

)+cos(

-

) )
which is minimum if
cos(

-

)=1
also bcox angle A=pi/4

+

=3pi/8
using this we will get the answer which we wanted that is
10sqroot(2+sqroot(2))/(2+sqrroot(2+sqroot(2)))
Isnt it a very good method!!!!!!
Plaese rate me bcoz i have really produced an excellent solution!!