Question

In triangle ABC angle A= 2 angle C. Show that c(b+c)=a².

Answer 1

In a triangle :  angle A =2 * angle C
         so angle B = 180 deg - 2 angle C - angle C
         B = 180 - 3 C

we know that
   Sin 2C = 2 Sin C Cos C
   Sin 3C = 3 Sin C - 4 Sin³ C  = Sin C [ 3 - 4 Sin² C ] = Sin C [ 2 Cos 2C + 1 ]

we know that:  a  / Sin A  = b / Sin B = c / Sin C
  =>  a / Sin 2C  =  b / Sin 3C  = c / Sin C
  =>  a = 2 c Cos C             -- (1)
 and  b = c {1 + 2 Cos 2C ]     --- (2)

Now find    a² - c² =  4 c² Cos² C - c²
                         = c² [ 4 Cos² C - 1 ]
                         = c² [ 2 Cos 2C + 1 ]
  substitute from (2),
   a² - c² =c² * b/c  = bc
  =>      a² = c (b+c)
proved.