Question

In given figure,AB=AC and BE and CF are bisector of angle B and angle C respectively.prove that triangle EBC = triangle FCB

Answer 1

let AC = x
then BC = 2–√2x

as AD and DB will be in ratio of AC and BC by angle bisector theorem

so let AD = y and  so DB = 2–√2y

but AD + DB = AB  = AC

so y+2–√2y = x
     y(2–√2 + 1) = x
   so y = x2–√+1x2+1

    on rationalising the denominator it results in 
      y = (2–√2 - 1) x

add x on both sides 
      y + x=  (2–√2 - 1) x + x
      AD + AC = (2–√2 - 1 + 1)  x
      AD + AC = (2–√2)  x
     so AC + AD = BC