In a scalene triangle `A B C ,D` is a point on the side `A B` such that `C D^2=A D * D B ,` sin `sin A * Sin B=sin^2C/2` then prove that CD is internal bisector of `/_Cdot`

Explanation

Let `angle ACD = alpha`
`rArr angle DCB = (C - alpha)`

Applying the sine rule of `Delta ACD and " in " Delta DCB` respectively, we
`(AD)/(sin alpha) = (CD)/(sin A) and (BD)/(sin (C - alpha)) = (CD)/(sin B)`
`rArr (AD.BD)/(sin alpha. sin (C - alpha)) = (CD^(2))/(sin A. sin B)`
`rArr (1)/(2) [cos (2 alpha - C) - cos C] = sin^(2).(C)/(2)`
`rArr (1)/(2) [cos (2alpha - C) -1 + 2 sin^(2).(C)/(2)] = sin^(2).(C)/(2)`
`rArr cos (2 alpha - C) = 1`
`rArr alpha = (C)/(2)`
Thus, CD is the internal angle bisector of anlge C